@Adam Marx Unfortunatley, most wholesalers won't allow any inspection period or contingency, that's kind of the risk of buying from them, but the benefit of getting a property under market value and without much competition.
The fact that you got in to see it is even pretty rare. The typical wholesaler is cash/hard money only, non refundable deposit, no inspection period and you pay both sides closing costs. If this scenario feels too risky, than you'll need to find your own off market deals.
If you are using a hard/private money lender that actually vets the deal (not just you), you might have some leverage. If they have a great track record and have worked with the wholesaler before, they may be able to negotiate an inspection or holdback, but don't plan on that as a strategy.
There are soooooo many wholesalers that took a 3 day course or learned it from youtube, it is likely they don't know what they are doing either. Explain your situation and that to keep the deal together in the best interest of you, them and the seller that this is not much of a request and lets get it done.
The tenant situation should have been resolved before you close. Once you close, they are your problem. You may want to make sure you have some extra money to give them cash for keys if needed.
Best of luck!