There are a couple of ways to bypass the Speaker, but they are pretty rare mechanisms.
Minority Leader Jeffries has signalled he may try a discharge petition. That requires a hard 218 votes. That would mean 6 Republicans if all Democrats vote (it isn't clear all Democrats would even vote for it given the Israel aid). It is a slow process that would take months to complete even if they got the 218 votes. The last successful discharge petition was in 2015.
Another option is to defeat the previous question. This only requires a simple majority not a hard threshold, so vacancies would lower the threshold. This is where a majority hijacks a rule and amends it for the bill they want to vote on. The last time this was successful was in 1988. This doesn't have the wait time that a discharge petition has. But as you can see the House doesn't like to do this.
Both are really rare. Mainly because House members don't want to defy party leadership.
They could also try to remove Speaker Johnson, but that's probably even more unlikely.
We'll have to see what happens. I know there is a majority of the House that supports Ukraine. But I don't know if there are enough Republicans that would defy Speaker Johnson or Trump for that matter.
Ukraine doesn't have the luxury of time here.
There are some means to get military aid to Ukraine outside of all of this, but I imagine that's a last resort for Biden that he would rather not have to pursue.