Hi Shawn,
I'm currently experiencing the same thing as I am set to close on a property that already has a current tenant on a month to month that doesn't meet my tenant qualifications. The process for the seller to evict them legally would be to send them a 15-day notice of termination (must be before the 15th of the month). However, with the COVID environment my attorney is advising all his clients against any evictions, as going to court could takes months to get them removed.
The main thing though regardless of if you get them out or not is to have the current seller and tenant complete a Tenant Estoppel form. A Tenant Estoppel is a letter where the Tenant confirms the details of their tenancy. This way after closing, there are no surprises. For example, let's say you need to evict in 6 months when you do go to Court, Tenant doesn't show up saying "I have a Security Deposit of $1,000.00" or "The previous owner told me I could stay here for a year" . . . or whatever craziness. The Tenant Estoppel is their certification of their lease terms. You can find these forms with a quick google search.
Hope this helps and good luck on the closing!