I figured out a solution to my problem. In case anyone else runs into a similar issue, here is what I did:
I created a query in my Access database that pulls the max value of my request ID, plus one. I added that as a secondary data source. On my submit button, I added a rule that runs a query (that secondary data source I just set up), then sets the request ID field's value to that value.
After doing that, I was STILL getting the primary key violation error on submit. I set up an autonumber on the Access table to use as the primary key, instead (just for testing), and allowed duplicate entries on my request ID field. STILL gave the PK violation. This time I knew something fishy was going on. I revisited my main data connection and modified it (went through the motions, didn't change anything, clicked submit). I noticed on the summary screen that it said submit was disabled due to a possible many-to-one relationship. I had a state abbreviation/name lookup table included in the main data connection as well. I realized I didn't need that table after all, deleted it, and all has been working fine ever since!
Long story short: I think all would've worked fine if I'd kept the request ID field as an autonumber and just removed the extra table from my main datasource. Still, I'm not a huge fan of autonumbers, and this solution is nice because I can use the request ID in the e-mail submission. If it was still an autonumber, the ID wouldn't be created until the form was submitted to the database.