Roussette In answer to your question, theoretically yes. But it's Congress as well as POTUS that can actually shut down the Govt.
I'm a retired Fed who went through this every year, multiple times, including three actual 'shutdowns'. Plus once as a retiree we left a Natl Park two days before a shutdown.
It works like this (simply put);
1-POTUS submits a budget proposal for the next Fiscal Year (FY) which begins October 1 to Congress for them to finalize and adopt by vote, then POTUS signs off.
2-Congress starts fighting about the proposed budget, each party wanting to add/subtract according to their own agenda. The clock starts ticking to 1/10 when the funding 'stops' (i.e. Govt 'runs out of money').
3-Congress can't get it's act together or get POTUS to sign off so it votes a 'Continuing Resolution' (CR) to fund the Govt at current budget levels for a certain time period (usually 90-120 days). POTUS signs off on CR and the Govt keeps running. This continues on with more CRs until Congress passes a budget and POTUS signs. I've known this to take until almost the following FY.
Here's where the fun starts
4-Congress can't get its act together or POTUS won't sign off AND someone gets a bug up their ass and says "We won't vote for (or sign off on) a CR until we get our way on so you better give it to us or we will shut down the Govt. When this happens the Govt effectively 'runs out of money' at midnight and 'non-essential services' shut down. Military, law enforcement, medical facilities, and many other Depts who are deemed 'essential' keep open and employees (usually a skeleton staff) must work with no promise of pay. Other employees are sent home on unpaid leave. My particular Dept was deemed essential after the first shutdown so we were required to work. But it's pretty well known that federal employees will get paid eventually.
It all works out in the end because Congresspeople really get shit from their constituency and shutdowns usually only last a few days until a compromise gets reached. But we're now dealing with a brand new animal. Trump doesn't give a shit how many people a shutdown hurts or for how long.
Makes me REALLY glad I'm retired.