In this case you'll need to accept an offer without an appraisal contingency.

If the vendor does not want an inspection, they should have declined the offer or countered.You could have your agent muse to the vendor that if they do not wish for an inspection, then you could offer 20-30% less and remove the contingency ;-)

Not even a counter offer.Plan C: Make an offer closer to asking price, but with a 90 day contingency to market the property and secure a tenant.

I need to find a foundation specialist who can look at this soon (before my inspection contingency expires) to determine how big of an issue it is and if it is possible to be repaired. 

It seems like the trend is to make offers waiving the inspection contingency or having a short inspection contingency period, which makes it almost impossible to get a good idea on what repairs are needed and how much it will cost before I'm committed to the deal.

The seller aka bank will permit the potential buyer to contact the local utilities and have them put utilities into the buyers name for one day for the inspection which of coarse is a contingency that you’d arrange as part of the sale .

-Have your agent write it as a contingency prior to closing, and if at all possible, have a contingency not to close until the tenant has moved out. 

For example is there a small but vocal contingent that wants to make a lot of costly changes? 

Our Financing Contingency ended 5/6, so now we are out a $5k Earnest Money deposit (plus the appraisal, survey and inspection, $7500 total) and don't get this great deal unless we act fast.In short, we provided to income docs for my wife, but our current primary residence is in her name, so that didn't help the DTI much, and we ended up losing the deal. 

10% of total renovation budget is a normal contingency balance.