There are two ways to go about this:

api1 = APIFunction["x" -> RepeatingElement["Integer"], Max[#d]&]
api2 = APIFunction["x" -> DelimitedSequence["Integer"], Max[#d]&]

and then

api1[{"x" -> "1", "x" -> "2", "x" -> "3", "x" -> "4"}]
api1[{"x" -> {"1", "2", "3", "4"}}]
api2["x" -> "1, 2, 3, 4"]

So the difference between RepeatingElement and DelimitedSequence is that the first takes repeated keys, while the second will split a string for you.

There is a much better way of doing that

APIFunction["d" -> RepeatingElement["Integer"], Max[#d] &]["d__json" -> "[1,2,3,4]"]

Look at the documentation of AllowedCloudParameterExtensions, to see why that works, and what else can be made to work.

Or another way of doing it is to use DelimitedSequence to define a JSON list parser for you:

APIFunction["d" -> DelimitedSequence["Integer", {"[", ",", "]"}], Max[#d] &]["d" -> "[1,2,3,4]"]