I don't know that the Defense Department has the authority, absent Congressional approval, to accept contributions of funds from private individuals or organizations for use in specific purposes such as funding military operations. Probably not.
However, politically it would be a way for people to support the troops and show up the Rats. The funds could be raised on the condition that if there is a legal obstacle to funding the troops, they would instead be contributed to Navy Relief or the Army equivalent, and therefore go to support our service families in another way.
Or better yet, if the funds cannot go to help the troops, they will go to fund campaigns against incumbents that did not support the troops.