Is there some time limit to sending the invoice, such that it is
invalid and the recipient is not obligated to pay it?
Yes, there is. The term you are looking for is Statute of limitations.
Basically, the statute of limitations sets the amount of time you have to collect money someone owes you. Usually, there is no specific law on when to send an invoice (with some exceptions, see below) - however, once the statute of limitations has expired, you are no longer required to pay it. On the flip side, even if an invoice takes unusually long to arrive (say a year instead of a few weeks as usual), but arrives before the statute of limitations expires, you must still pay it - there is generally no right to receive a "timely" invoice.
In the United States specifically, the statute of limitations for civil cases (such as contract disputes, as in your case) will depend on the state. It is typically between two and six years.
On findlaw.com, there is an overview for all states. It looks well-researched, though I cannot vouch for the correctness.
Some details about the rules for invoices:
There is no general law (I could find) that requires an invoice to be sent within a certain time frame. However, there are laws for some specific cases. For example, credit card companies must mail or deliver credit card bills at least 21 days before payment is due (WHAT YOU NEED TO KNOW: New Credit Card Rules, by Federal Reserve).