Yes.
I've seen several residents from my own program get into great fellowships who were IMGs, though my program was not low ranked, more on the order of medium ranked.
The competition field for psychiatry fellowship tends to greatly drop vs residency.
Now of course this may change given that this year there was an upsurge with the amount of people going into psychaitry residency--which may create a delayed effect where fellowship may get harder & more competitive in 4 years time.
If so...how would one do that?
Well some programs literally don't fill up their spots. This includes some of the top fellowship programs. Several fellowships I've seen only have a handful of applicants (like 0-10) for 1 spot vs dozens to hundreds for 1 spot in residency.
Just do well in the exams you take during residency, and impress the hell out of your attendings in order to get good recommendations and such?
Yes, given that there's less applicants, fellowships are more willing to look at the person on an individualistic level, rather than filter everyone out via board scores, medschool grades & such. They didn't even look at PRITE scores, though if you do well on the PRITE, you might want to mention that in your applications to impress them.
Do they take into account your medical school performance?
I only saw one place do that out of about 10 programs, and the interviewer was really trying to do everything in his power to make me feel uncomfortable during the interview, which pretty much didn't faze me because I already got into a few programs that I wanted to be in more than that one, and have seen that type of thing too many times to care. The guy tried to pick at everything in my record, started asking me what I actually did during my vacations in a very condescending manner. Hey, fault me for being a human being who wants to have a mentor that I actually get along with. I know being a doctor I'm not supposed to have emotions & I'm supposed go for the guy with the highest prestige who treats me like a servant.......
I've known about a dozen people who got into fellowship, and none of them had their USMLE scores, or medschool scores brought up. The more important things were letters of recommendation & things you may have accomplished in residency such as publications, and work experience in the specific field you are applying into.