Do I need a conference paper to vouch for a tool I built in my undergrad when applying for a PhD?
Having published a paper does have a positive impact on PhD applications. (But you don't "need" one in the strict sense - it is quite possible to get into grad school without having published.) But the reason is not because it "vouches for" your tool working. In most cases, the reviewers never get to try to execute your code, so it can hardly speak to that.

It does, however, indicate your ability to produce something novel, of interest, and to describe it in a manner that receives the approval of other scientists in the area - in other words, your ability to do research. Which is exactly what the PhD admissions committee will be looking for.

Whether you think that is worth the conference costs is a matter of personal judgement - but you should first look for other avenues of funding, such as your advisor or a travel grant from your university or the conference or a third party promoting undergraduate research etc.