If there is a contract with a government and there is no witness to it, what impact will it have?
Is the contract valid?
Contracts are valid when two parties come to an agreement; witnesses are not strictly necessary.
The purpose of a witness is to state: so-and-so signed (or did not sign) this particular document at such-and-such time on such-and-such date, in case this fact is later disputed.
Generally, with a modern government contract, a witness may or may not be necessary: most drafting and tendering rules, to say nothing of modern paper/email trails, are enough to prove in a court of law the document signed was reflective of the "meeting of the minds" required to find that a contract has been made.
Personally, the only time I have ever been required to have a witness certify any document that I have signed has been when I accepted an insurance payout for a car accident (and then, it was less the money and more the fact that I was signing away my ability to sue the other party).
Establishing a contract requires only an offer, acceptance, and an exchange of value between the parties.
It is rare that a contract even needs to be written, and I know of no circumstances under which a contract must be witnessed by a third party.
The main impact associated with a lack of witnesses is that it becomes harder to prove that the parties reached an agreement, but the courts deal with this sort of thing all the time. From what I've seen, the fact that one party started to comply with her contract obligations might alone be enough to persuade a court that there was a contract.