While in principle everything that you use in your research should be paid for by those funding your research, in practice it's not unusual to buy oneself small supplies at the "minor office supply" level for sake of expedience, or because they're outside of the main line of one's research.
For example, a student might want a $10 piece of electronics today that would take several weeks to acquire through normal channels, and then might decide it's not worth it to them to get reimbursed. You might also buy something for yourself if you want to be the owner: for example, I bought my own laptop late in graduate school, rather than using a school laptop, because I wanted it to be my own personal machine that I could take with me when I finished.
What you have written, however, sounds like a much more problematic situation: it appears that you're effectively being asked to lie to your professor's funder, by misrepresenting how their money is being spent. If it's a significant amount of money, then you could be abetting in a felony-level crime. As such, if it is a significant amount of money, I would strongly advise you to not buy the equipment yourself. Instead, I think you should tell your advisor that you need the equipment, but if they don't get it, you'll just have to do your best without.
If your advisor doesn't buy the equipment, you shouldn't badmouth your advisor to the funder. Just do your best, and if they ask you why you don't have the equipment you need, you can tell them the honest truth: you've asked your advisor for it and are hoping you'll get it.