It seems they want to issue you a 1099 form (probably a 1099-MISC). That means that either they think the grant might be taxable income to you, or at least, they are taking no chances with their tax return and leaving the tax issue up to you.
Whether this is some form of taxable income or a tax-free gift depends on what you have to do to get the money. We would need more details to think about that.
If you get a 1099-MISC, but you think the money should not be taxable, there is a way to zero out the 1099 on your tax return.
If they are giving you a no-strings gift, then it doesn't matter (to the IRS) what you do with the money or when, unless you have a contract with them that requires you do something. If the money is taxable, you pay tax for the year when you receive the money, even if you don't spend it. If it is not taxable, but you get a 1099, then it is reported and zeroed out for the year when you actually get the payment.
Lastly, if you pay medical bills with money that you did not pay tax on, you can't take a tax deduction for those medical expenses.
