It would be possible, but here’s why you wouldn’t want to do it.
Firstly, inflation, classical inflation targets are set at around 2%. That represents the amount of money that is printed each year to pay for the additional spending not covered by the taxes raised.
If you replace taxes with money printing, that number would rise to something like 60 – 80%
This assumes you don’t tax anything, no income tax, corporation tax, taxes to pay for pensions, health care, welfare, military, everything.
In reality running a country, any country is expensive and most of the work you do as an individual is towards that end.
That would be a hard sell for any government to convince their electorate.
But you still wouldn’t want to do it because hyperinflation would kick in and you would be regularly changing your monetary base unit accounting system, unless you’re happy paying a billion dollars for a loaf of bread.
That’s not the main problem however. The U.S. dollar is the global reserve currency, by printing money to pay for government spending, you’d be “taxing” all holders of the U.S. dollar. Foreign nation states, corporations and private holders.
This would not be tolerated and the dollar would be abandoned very rapidly, which is what’s happening, but slowly.
Taxing people within your borders and your citizens outside of those borders, ensures only people benefiting from government services contribute to it financially.