If you have agents involved then get their brokers to look at the contract.
Contact a real estate attorney.
The paying above appraisal is something that is being seen more and more often among homes that are private investor owned (flipped) and that the same buyers have offered more than what it is listed for. The verbage usually reads something like this ( pay UPTO $5000.00 over appraised value NOT to EXCEED $160,000.00).
The only part I don't understand is why the seller would need to sign for a second appraisal. What do they have to do with it?
You need to speak to your agent and lender. Get your agents broker involved to sort this out.
There are so many questions that come to me regarding your question.
1.) Buyer's normally pay for an appraisal through their applied lender. If the seller is financing, it is unclear who wants the second appraisal. There are situations where the lender will request two appraisals. Sometimes with FHA at 3.5% down. Sometimes when the buyer owns more that 4 properties.
2.) I don't understand the 5K clause. If the seller paid for the inspections and appraisal, I might be able to see what there was a reimbursement if it didn't appraise.
3.) To know what you can do, you need to sit down with your real estate agent and review the contract. The only way to know your options is to have a complete understanding of the contract. If you don't have a real estate agent, a real estate attorney will be the next option.
To many guesses could cause more issues about what can be done to satisfy all with contract information given.
I agree with Sara, what do you need a second appraisel for? unless the bank wants to order one. This sounds strange to me. if you wrote that you would pay UP to $5000 to cover if it did not appraise for the full 160k, then there shouldn't be a problem, but if there is..... some data is missing as to what EXACTLY was written in the contract. Your Agent should be all over this,if not call his broker, if that doesn't work or aren't using a seperate agent or any agent ( not smart) then contact an attorney before you do anything else. This sounds very strange something is missing here, have someone (attorney) look over this contract as it should be crystal clear what the $5000 was suppose to cover and nothing else and not open to interpetation by the seller.
Good luck. You may want to hire a lawyer if you believe that you are in the right.