As I understand it, this is the supplemental expenditures bill. So routine expenditures (e.g., troop salaries) are taken care of out of the budget passed at the beginning of the year.
So nothing dramatic happens after veto. The troops don't go unpaid. We don't run out of ammo. The troops don't come home. Nothing.
What happens is that money gradually runs out. Projects don't get started. Then the administration still has flexibility to shift around spending priorities. They can cut civilian staff in Okinawa and not stock the PX at Patuxent River to shift around funds. Meanwhile, events continue to unfold and the military labors at an increasing disadvantage.
Eventually, the military runs out of fancy tricks and supplies get tighter, but there is a lot of slush to maintain the status quo.
To kill the war, as Congress did in 1975, it would have to actively "defund it," as we say in Washington. That is, Congress prohibits the federal governmnt from spending money on something or -- in the case of an agency -- Congress gives it a line-item budget of zero.
So yes, this is an important showdown. But it ain't a Constitutional crisis.