Must the bride be present at the marriage contract?

It is not required as a pillar; the contract takes place between the wali and the groom before two witnesses.
The bride is not the party who pronounces the offer (ijab) in the Shafi'i school, so her presence is not a condition. However, her consent must be obtained first, and in Malaysia she usually has to sign a consent document and is questioned by the registrar. Most brides attend because of custom and for the paperwork. See Marriage contract.