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That's usually the job of a realtor that handles the rental listing. And then of course, the full service would be to handle the listing, screen and make recommendations and collect payments for you.....but, I suspect you are trying to save that expense, which is understandable.
Do you have a mortgage loan officer that you work with that would be willing to pull their credit and review a basic rental application? I traditionally do that for my clients as a courtesy, especially if I did the new loan where they now live or the investment loan for the rental.
Seriously, ask, most loan officers really do not mind. Just make sure the rental application gives them permission to discuss their qualifications with you.