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How do I redirect output to a variable file name in a for loop?

This doesn't work:

for ((j=501; j <=550; j++)); do 
curl -x http://us-wa.proxymesh.com:31280 "https://catalog.loc.gov/vwebv/search?searchArg=D$j&searchCode=CALL%2B&searchType=1&limitTo=none&fromYear=&toYear=&limitTo=LOCA%3Dall&limitTo=PLAC%3Dall&limitTo=TYPE%3Dall&limitTo=LANG%3Dall&recCount=1000" >trial_$j_out.html 2> trial_$j_error.txt;
done

I keep on having trial_.html and trial_.txt written.

As pointed out in comments, _ is a valid character in a variable name. The shell will interpret trial_$j_out.html and trial_$j_error.txt as consisting of text surrounding the variables $j_out and $j_error.

Use $j instead, just like you (needlessly) did in the actual URL:

for (( j=501; j<=550; j++ )); do 
 curl -x http://us-wa.proxymesh.com:31280 
 "https://catalog.loc.gov/vwebv/search?searchArg=D$j&searchCode=CALL%2B&searchType=1&limitTo=none&fromYear=&toYear=&limitTo=LOCA%3Dall&limitTo=PLAC%3Dall&limitTo=TYPE%3Dall&limitTo=LANG%3Dall&recCount=1000" 
 >"trial_$j_out.html" 2>"trial_$j_error.txt"
done

This will properly delimit the name of the variable from the following _. The $j in the URL does not need curly braces as the character & is not valid as part of a variable name.