You could proceed like this:

image = RandomImage[{0, 1}, {5, 3}];(* Just test data *)

ScientificForm allows you to format number using some version of scientific notation along with your own customizations.

ScientificForm[0.00962947, NumberFormat -> (StringJoin[#1, "*^", #3] & )]
(* 9.62947*^-3 *)

While the above looks like plain text, it's actually a special front end "object":

FullForm[ScientificForm[0.009629474952816963`, NumberFormat -> (StringJoin[#1, "*^", #3] & )]]
(* ScientificForm[0.009629474952816963`,Rule[NumberFormat,Function[StringJoin[Slot[1],"*^",Slot[3]]]]] *)

So, before exporting, you need to turn it into an actual string.

ToString[ScientificForm[0.009629474952816963`, NumberFormat -> (StringJoin[#1, "*^", #3] & )]]
(* "9.62947*^-3" *)

Now we can turn that into a function that process each pixel value of our image.

readyToExport =
  Map[ToString[ScientificForm[#, NumberFormat -> (StringJoin[#1, "*^", #3] & )]] &, ImageData[image], {-1}]
(* {{"5.80547*^-1", "7.10634*^-2", "1.66409*^-1", "7.99704*^-1", "7.71393*^-1"}, 
    {"1.45017*^-1", "2.94124*^-1", "5.34167*^-1", "8.0723*^-1", "6.22651*^-3"}, 
    {"1.41292*^-1", "9.58796*^-1", "3.46061*^-1", "3.44042*^-1", "8.9646*^-1"}} *)

Now we can export. I'll use ExportString to demonstrate, but you can just use Export and supply a filename.

ExportString[readyToExport, "TSV", "IncludeQuotingCharacter" -> False]
(* 9.3502*^-1   4.1657*^-1    3.67851*^-1    3.34613*^-1    7.63106*^-1
   6.01477*^-1  6.27864*^-1  5.98659*^-1  3.99904*^-1  7.38507*^-1
   5.9335*^-1   3.01261*^-1   9.08788*^-1   9.16284*^-1   9.36047*^-1 *)

If you really need just spaces rather than tabs, then you'll need to do a bit more preprocessing.

Yes. I can use ImageData to generate an output text file, but it is littered with outer brackets, inner brackets, and extraneous descriptors that other image processing codes (e.g. ImageJ) don't like. I would like to just dump 32 rows with 50 delimited numbers per row.

Have you tried ImageData?