The big box store can't make you do anything. But they can decide to only accept offers from general contractors who agree to pay for background checks on their employees and subcontractors. As a general contractor, you are not being hired as an employee under the control of the big box store, but rather you are an independent contractor. Thus, you make an offer to the store and hope that they accept yours out of all the offers/quotes they receive. Even if the big box store has made it clear that they expect whoever the GC is to pay for background checks of its employees and subcontractors, you don't have to promise to do this, but if another general contractor agrees to do it, you may be lose the contract.
However, if you've already been hired as the GC, never promised to do this in your original offer, and now the big box store is bringing this up after the fact, then they are trying to add a new term to the contract, and you probably don't have to do it. And if the big box store backtracks because of it, they they will likely be breaking your contract. But it all depends on the contract both sides agreed to, if in fact you have agreed to something.
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If the requirement is part of the contract you have to comply. If it is not part of the contract and it is signed and in force you do not. If the contract has not been signed it is up to you if you wish to comply and accept the terms.
If that is a term of your contract, then yes, they can make you pay those costs.
If you already have a signed contract and that is not one of the provisions, then they cannot force you to do it. Background checks do not usually cost that much. Unless you have hundreds of employees, it should not be that expensive. Perhaps you could limit the employees that work on that project and get the store to agree to checks on just those employees.
As for your subs, have them pay for the background checks for their employees as a condition of your contract with them.
Yes, if this requirement is part of your signed contract with the store. You are now contracually obligated to perform. However, if your contract does not contain this provision and has an "integration clause" that says, essentially, that the written contract encompasses all agreements between the parties on the matter, then you have a solid argument you do not have to pay. In latter case or if you have not yet signed the contract, you can refuse this clause, negotiate to split the cost with the store or get another benefit for taking on the cost.
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