Qfyilyi

If we wanted to create a rule for the government budget that said that would prevent government from NOT leaving the government funded, would it require a constitutional amendment or could/should this be done by an appropriations law requiring only a majority of legislators?

Here's an example of what the wording might be like (not very formal, but I hope you get the idea of what I mean):

If the House of Representatives or Senate fail to agree on new funding
or the Executive Branch fails to approve a budget, then the previous
funding shall stay in effect until a new one is agreed on. The
government or no portion of it may be shut down without funding as a
result of the expiration of a budgeting law.

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.

A normal vote is all that is needed.

A constitutional amendment is needed to prevent it from being possible.

The house originates the bill. It then goes to the Senate for changes/approval. Once the House and Senate have a bill to pass, they send it to the President to sign (bill is law) or veto (goes back). If it goes back then the bill requires a 2/3 majority to override the veto. To prevent it from happening, an amendment would be needed to alter how the separation of powers between legislative and executive function. That is, a clause needs to be added or altered. A law alone cannot circumvent a veto or required 2/3 override.

In this (current) case, it is the same as a veto. Two-thirds majority will overrule the veto and pass the budget. Additionally, the lack of the President's signature doesn't actually shut the government down, it only prevents it from spending more than what it is currently allowed to spend.

The shut down occurs when the various departments run out of money they were previously allocated.

Section 7 here.

To make changes that would prevent the government from ever shutting down would require amendments to the constitution. Specifically areas citing presidential approval or new language supporting a new mechanism to keep the budget operating.

To address your idea, this is already somewhat done in the sense that all essential federal government employees continue to work without pay. Once the budget is approved they will receive their back pay.

Edits to make improvements. Edit: To include information from comments