In the same way you made customers and track their balance, you can make a "payments" table, check cash etc, then your grants and track the balance in there.

Or you can make a new "item" with a negative cost and put it in the order details on invoices if you want to do it that way.

Or you can make a table that tracks the total amount of each grant and use VBA to remove some of the balance when you use it on an invoice.

Lots of options, I'm not sure what suits your needs best.

If you're looking for videos it looks like Access Dev 8 cover partial invoice payments

and Access Dev7

You might also check out the Access Payables Seminar to see if that would be what you need.
There is a free sample db you can download before you buy.  Also I'm sure Richard would answer questions you might have to make sure it would work for your needs.

Do a search for Many-to-Many and Relationships.

Sounds like you definitely have a many-to-many situation going on.  
One grant can pay many invoices / line items and one invoice can have many payments.

There are so many different ways to track this, and it's really up to how YOU want to do it. I would make a GrantT with the grant details, and then it could have Orders as child records. Easy to tally up too.