Hypothetically speaking, if an incoming Attorney General wished to stop an ongoing investigation without cancelling it and tried to make the investigation grind to a halt by removing funding, would it be possible for the House to fund the investigation if they wished to see it completed?
The House of Representatives has its own budget. It determines how that budget is spent, although the budget itself has to be approved in a budget bill with Senate approval and either presidential acquiescence or a veto override. Within that budget, it could fund an investigation.
That's not really necessary though. The House also has the ability, even the duty, of providing oversight. It can investigate the president directly. And unlike a special counsel, its investigation would be mostly public.
In the specific current example of the Mueller investigation, cutting funding without ending the investigation just leaves the problem open. I suspect that it is more likely that they will tell Mueller to issue a report that can be published. The reason is that so far Mueller does not seem to have any evidence on the president that he is supposedly investigating. They want that to be published. They will then use that against any future House investigation. They will call the investigation a witch hunt after Mueller's investigation cleared the president. They will do so even if Mueller disagrees with that characterization. This is speculative, but entirely consistent with their previous behavior and statements.
Another step that they can take now that Jeff Sessions is no longer in the way is to start investigating Democratic politicians and leaking information about them. Sessions wouldn't sign off on the "Lock her up" investigation of Hillary Clinton, but the acting Attorney General probably will.