It might not be an option for you but in my case I display the form embedded in a website where the user is logged in, so it’s possible include their name in the embed url to prefill a ‘Submitted by’ field.
I’ve used ‘lookups’ successfully to have people select their name or add a new record and I make the field required on the form. But… it does mean people can search alllll the names in the lookup.
If you have Zapier, you can have it automatically look for blank username fields and then backfill the field with the creator from the field history. But, since this is coming anonymously from a form I doubt it would have a username to lookup. If you have people ‘type’ their name or email and have a system already containing that info (like your email contacts list) you could potentially make it work.