With a deposit and a mutually agreeable price, you should be able to get an accepted offer. When you present yourself before the board for approval, they will approve or reject you based primarily on your credit, debt, and reserves, not on if you received a grant.
It is always a great idea to apply for grants whenever possible so long as they do not impede your ability to close. Your realtor, loan officer, and attorney should work together to make sure you can achieve both. Please let me know if you have any questions.