According to the Constitution, Congress has the power:
To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
So, without a constitutional amendment, there is no way to fund the military without passing at least a military spending resolution at least once every two years. But this also implies that for non-military spending, Congress has the power to pass a budget that lasts more than two years.
I think that Congress could, without a Constitutional amendment, specify that a previous budget would be in effect if no new budget was passed - at least for non-military spending. There's nothing in the Constitution to prevent it. However, you wouldn't want to do this for everything in the budget.
Let's say the budget one year contains $50 million to build a particular bridge in a particular location. If this provision kicked in, then another $50 million to build the same bridge would automatically be in the next year's budget. Are we going to build a second bridge in the same location? Or maybe there's $20 million to upgrade a particular agency's computer system. Are we going to upgrade it again? Probably not.
And things might become clouded when it comes to how the government is often funded - through continuing resolutions, instead of a budget. If a particular department gets $100 million per year in a budget, and they get $75 million for the first 9 months and then $30 million in a 3 month continuing resolution, how much should they get if this provision kicks in? $105 million, which is what they got in the last 12 months? $100 million, which was what they were allocated in the last actual budget? Or $120 million, which is the extrapolation of the last continuing resolution?
Congress could, of course, clear up any ambiguities by wording the laws correctly. And they could specify when an item was a one-off that shouldn't be automatically renewed. But they still might not want to pass this sort of law. Perhaps one reason is that it would be too easy to fall back on this, instead of passing a good budget that reflects the needs of the current year instead of the previous year.