First note that it's a question of institutional policy whether human subjects research that is not federally funded requires IRB review. The common rule requires that federally sponsored research receive IRB approval but does not require institutions to do IRB review for research not funded by the government. However, many universities require that all research involving human subjects receive IRB approval whether or not it is funded by the federal government.
Assuming that IRB review is required, it is critical that you not begin the actual research until after IRB approval has been obtained- starting without approval is a big no no.
Normally, IRB approval must be obtained before your institution enters into a contract with the funder or the grant is awarded. For example, at our institution, the restricted funds accounting office won't do the paperwork for a research contract involving human subjects research until the IRB approval has been granted. We are also not allowed to submit a proposal to a funding agency without (edited to clarify this) submitting the proposal to the IRB for approval. There's a good chance that your institution has a similar policy.
Even if your institution doesn't have a specific policy about this, you should not try to get your institution to enter into a contract before the IRB approval is granted. If for whatever reason the IRB approval is denied, then you would have gotten your institution into a contract that it can't fulfill.
If I was asked to take on a research contract involving human subjects and wanted to accept the funding, I would tell the funder, "Yes, I would like to accept this funding. However, I will have to get IRB approval (as well as other approvals) before the institution will enter into a formal contract."