I’m looking at a job application that asks to post “last 3 employers.” Would I be able to include independent contracting work and self employment?
Yes, you can include contracting work and being self employed. However, the question asks for your last three employers, so whatever you list should be your most previous three employers without gaps.
The recruiter/hiring manager will also see your resume, which you can use to highlight the most important previous roles, even if not the most recent.
It's up to you whether you aggregate freelancing/self-employed work together into a single item or break it up into additional details - do what you think enables you to list the recent roles you're most excited to share.
I've been a contractor, both W2 and corp-to-corp (aka self-employed), for nearly 30 years now. When I list them on my resume, what I put depends on the type of job. I nearly always put the company I was contracted to because it's more likely to be a company that is recognized instead of the staffing/consulting company that actually pays. me. I will then often put "(via SuperRecruiters)" where the silly name I put there is actually the company that paid me. My current situation is even more complicated. I put the Fortune 50 company I worked for for 3.5 years and put (Via UberConsultants) but they're not the ones that paid me. It can all get very complicated but the bottom line is that you want to put where you did the work but you can also put who employed you, because even if your work-quality reference isn't from them, they'll be the ones to verify your employment for a background check.
In the cases where I'm self-employed, I just put the name of the company I was working for and merely put something like (contract) in parenthesis.
The above info is for resumes in particular but can be used for applications. BUT if you've already submitted your resume and this is just for either a background check or following an HR procedure, put the people that paid you or "self-employed" and leave it at that. For a background check you might get the BC company calling you asking for whom you did work, but that's all.