That looks like an error from a pre-release build of 13.2. You can work around it by passing a named object, like CloudObject["myip.api"], as the 2nd argument to CloudDeploy:

CloudDeploy[Delayed[$RequesterAddress], CloudObject["myip.api"]]

You may also be able to fix the problem by calling Needs["UUID`"].

I changed the code to remove the Delayed and it worked successfully:

CloudPublish[
 FormFunction[{"city" -> <|"Interpreter" -> "City", 
     "Input" :> $GeoLocationCity["Name"]|>}, 
  Grid[{{Style[#city["Name"], "Title"]}, {Style["Current Temperature",
        "Subsubsection"], 
      IconData["AirTemperature", WeatherData[#city, "Temperature"]], 
      QuantityForm[WeatherData[#city, "Temperature"], 
       "Abbreviation"]}, {Style["Map", "Subsubsection"], 
      GeoGraphics[EntityValue[#city, "Position"], 
       GeoBackground -> "VectorClassic", ImageSize -> Large]}, {Style[
       "Image", "Subsubsection"], 
      GeoImage[EntityValue[#city, "Position"], 
       ImageSize -> Large]}, {Style["High Resolution Image", 
       "Subsubsection"], 
      GeoImage[EntityValue[#city, "Position"], GeoZoomLevel -> 15, 
       ImageSize -> Large]}}, Alignment -> Left] &, "CloudCDF"], 
 Permissions -> "Public"]

Here's some pictures.

Awesome, nice to see this!

The extra Delayed shouldn't be necessary, if that is necessary to make it work then that is curious.

I now have a simple API that lets you enter a city and see basic weather data

CloudDeploy[FormFunction @@ 
  APIFunction[ "location" -> "ComputedLocation", 
   Grid[{{Style["Current Temperature", "Subsubsection"], 
       IconData["AirTemperature", 
        WeatherData[#location, "Temperature"]], 
       QuantityForm[WeatherData[#location, "Temperature"], 
        "Abbreviation"]}, {Style["Current Relative Humidity", 
        "Subsubsection"], 
       IconData["RelativeHumidity", 
        WeatherData[#location, "Humidity"]], 
       QuantityForm[WeatherData[#location, "Humidity"], 
        "Abbreviation"]}, {Style["Current Wind Speed", 
        "Subsubsection"], 
       IconData["WindSpeed", WeatherData[#location, "WindSpeed"]], 
       QuantityForm[WeatherData[#location, "WindSpeed"], 
        "Abbreviation"]}, {Style["Current Wind Direction", 
        "Subsubsection"], 
       IconData["WindDirection", 
        WeatherData[#location, "WindDirection"]], 
       QuantityForm[WeatherData[#location, "WindDirection"], 
        "Abbreviation"]}, {Style["Geographical position", 
        "Subsubsection"], FromDMS[#location], 
       DMSString[#location, "Position"]}, {Style["Current time", 
        "Subsubsection"], 
       DateString[Now, "DateTime"]}, {Style[
        "Geographical xyz position", "Subsubsection"], 
       GeoPositionXYZ[#location]}, {Style[
        "Geographical projected position with Bonne grid projection", 
        "Subsubsection"], 
       First[GeoGridPosition[#location, "Bonne"]]}}, 
     Alignment -> Left] & , "CloudCDF"], "versa"]

I was able to change the map with some code:

CloudDeploy[
 Delayed[{LatitudeLongitude[$GeoLocation], 
   GeoGraphics[$GeoLocation, GeoRange -> Quantity[50, "Kilometers"], 
    ImageSize -> Large, GeoBackground -> "VectorClassic"]}], 
 CloudObject["myip.api"], Permissions -> "Public"]

I would also like to option that lets the user override the default city. I'm not sure if an api can have intereaction with a user or if the right function would be FormFunction to ask the user for a city and then give data but it seems like there should be a way to add manually overriding and defaults.

I don't really understand all the HTTP material, but is there is a simple way to make a form like the one Hannah created that sets the default to the current city? I think I'll wait for a programmatic API way but is there a way for a user to open the cloud deployed webpage and automatically see their city and then if they want they can search for a different city? I've tried using Delayed with Hannah's code but I haven't been able to make it change from Huntington to the location of the VPN server.

Is there a way to make a programmatic API with a default that could be executed with for example HTTPRequest in Mathematica that defaults to the API's Geolocation if no parameters are specified and otherwise if a location is specified it uses that instead?

What is "the API's geolocation"? Do you mean the geolocation where the servers run? You might think you could discover the cloud's location by making an HTTP from within the cloud to itself, but the networking there is going to use a private IP address (aka RFC 1918 address), one that doesn't have a GeoIP mapping. We do have a notion of a default geolocation for the server, but I don't think we expose it in a WL variable, it gets assigned within the web server and then passed to WL. We could do that, and there is precedence for things like that, e.g. $SystemTimeZone is the time zone the computer has before a user (or the cloud) started localizing it.

The server uses the requesters IP address to decide which city to compute data for.

Yes, the geo location (lat/long), country, and city are determined in cloud by using a geo IP mapping that determines location from a public IP address. If the requester IP address is not a public address, there is no mapping, so the cloud sets these to a default location/country/city, which is that of the Wolfram company HQ in Champaign, Illinois for public cloud. For private clouds the default location is configurable.

I know the server has a static IP address but if people from different parts of the world queried the Wolfram Cloud API could it tell where they were?

The server static IP address doesn't enter in to the discussion at all. It tells where they come from using the IP and the geoIP mapping described above.

If things aren't working as expected in your testing with a VPN, you'd want to confirm that the request to the cloud is actually going through your VPN, and you'd want to see what requester IP address the cloud actually sees, and confirm that's what you expect.

I created a simple function to retrieve weather data for a city.

CloudPublish[
 APIFunction[{"city" -> "City" :> $GeoLocationCity}, 
  DeleteMissing[
    AssociationMap[
     Function[property, 
      WeatherData[First@GeoNearest["WeatherStation", #city], 
       property]], {"CloudCoverFraction", "DewPoint", "Humidity", 
      "Pressure", "Temperature", "Visibility", "WindChill", 
      "WindDirection", "WindSpeed"}]] &], Permissions -> "Public"]

I can use the function with a default city:

URLExecute[
 HTTPRequest[CloudObject[
  "https://www.wolframcloud.com/obj/17846929-623c-4a7a-8172-\
9837ec51a19d"], <|"Query" -> {}|>]]

The function returns the following:

<|"CloudCoverFraction" -> 0, 
 "DewPoint" -> Quantity[-2.19999, "DegreesCelsius"], 
 "Humidity" -> 0.305, "Pressure" -> Quantity[1034.1, "Millibars"], 
 "Temperature" -> Quantity[15, "DegreesCelsius"], 
 "Visibility" -> Quantity[16.09, "Kilometers"], 
 "WindChill" -> Quantity[14.11, "DegreesCelsius"], 
 "WindDirection" -> Quantity[30, "AngularDegrees"], 
 "WindSpeed" -> Quantity[14.76, ("Kilometers")/("Hours")]|>

The API function also supports manual inputs. For example, the following code will return data for New York City:

URLExecute[
 HTTPRequest[
  CloudObject[
   "https://www.wolframcloud.com/obj/17846929-623c-4a7a-8172-\
9837ec51a19d"], <|"Query" -> {"city" -> "New York City"}|>]]

Here is the output:

<|"CloudCoverFraction" -> 0, 
 "DewPoint" -> Quantity[-8.29999, "DegreesCelsius"], 
 "Humidity" -> 0.248, "Pressure" -> Quantity[1035.8, "Millibars"], 
 "Temperature" -> Quantity[11.1, "DegreesCelsius"], 
 "Visibility" -> Quantity[16.09, "Kilometers"], 
 "WindChill" -> Quantity[8.93, "DegreesCelsius"], 
 "WindDirection" -> Quantity[40, "AngularDegrees"], 
 "WindSpeed" -> Quantity[18.36, ("Kilometers")/("Hours")]|>

I also made a form that displays information for a city like a city's area and median age. I also made an API function to call from Mathematica at CloudObject["https://www.wolframcloud.com/obj/73c731cb-7afb-4b8b-ad06-509c3aab3dd1"]