I never knew that money would be such a motivator of family problems, but what should I do?

Yep, always keep records. My Moms bank statement showed everything going in and out. If I bought something out of pocket, I wrote myself a check once a month and put the receipts in an envelope with the check # on it. Keep timesheets for the aide with her signing off on it.

Actually the BIL has no rights. So I may tell him to mind his own business. If he is so worried about his MIL, he can care for her, If he threatens with APS tell him to go ahead and call. You have nothing to hide. APS may be a good thing. Are you POA, if so you are not obligated to talk to ur Sister about Moms finances. Better u don't.

Turn the tables on him. I wouldn't apologize if you feel u did nothing wrong. And keep a copy of that letter. May need it to show to a lawyer if he keeps harassing you and it is harassment.

So the letter he wrote to you is by "snail-mail"?
And he is basically accusing you of fraud?
Does your Sister know that he wrote this letter?
In what way does your sister's husband say you mistreated your sister? For what are you to apologize?
I believe I would scan this letter, and the envelope it came in. File it. Should any accusations go out to the family via email I would make a copy of this letter and as RR said, reply to all. Post it for all to see.
Are you the financial POA for your Mom?
She is living independently? She is competent? She is handling her check payments for her care, her groceries?
I am guessing that when she sends you out with a storelist you are using her money and you are keeping the receipt and being compensated for what you spent? I am assuming you are keeping files?
I do know the cost of a few cases of boost, and YIKES!!! Bet it's even worse with inflation.
I am finding stuff here is missing and there are gaps as was noted by a few other respondents.
I can only say that you should keep good records and good files of any expenditures. Keep a daily diary of all you do with, for Mom including all expenditure, how made, how compensated for.
It seems from all you say that you are the ON SITE (or near to site) caregiver. I don't understand the questioning about money, because putting Mom into ALF would be so much MORE money than she is spending now with your help.
I would sit down with my Sister, to be honest, on her next visit. I would ask what is up with hubby and does she share his feelings? What are they suspicious about? What can you answer to alleviate any suspicion. If BIL has a bee in his bonnet you could have APS on your doorstep next, in all truth. We see this happen.
This is all quite sad, but I cannot tell what in the world has led up to it in a situation that seemed/seems to be "under control".
I wish you the best. If you get answers I hope you will update us.
There is so much I feel I am missing.

I would accidentally reply to all bcc to his threats and intimidation.

That way everyone will expect his nasty email and already know what he said.

Oops, my bad, I just have such a hard time with technology.

Is your mom mentally competent to deal with her own finances?

CM, kudos to you for raising the issue of the gap.   I wondered about the complaints by the husband, but that happens so often I just thought that perhaps he was one of the meddling types.

You put the narrative in perspective, as always.

Mmm. I think there's a gap in your narrative, isn't there.

So your brother in law wrote to you alleging that you have stolen from your mother and demanding that you apologize to your sister.

Apologize to her for what? That's what the gap is. What did you say to her?

Anyway - so basically, you've bought items your mother needed, your sister wants to know where the money's gone, you either can't or won't show her the receipts (I hope to goodness you have actually got receipts, or at least some record of what you've spent), she became suspicious, you became offended, and the whole thing descended into handbags at dawn. Did you and she also disagree about what kind of thing your mother needed?

Proportion and perspective will help all of you, here. Is the dispute about fluffy blankets and new slippers, or have you remodelled the basement and bought a car?

PS Even if I don't sound sympathetic I honestly *am.* I don't speak to my controlling sister any more, and I nearly decked my sister in law. But I don't want those feelings to get worse for you!

First, document ALL your contacts with your sister and husband.  You may need that data  in the future for any potential legal action.

Second, it's hard to determine whether or not husband is attempting to scare you, has no control over his emotions and interactions, and/or has no ability or knowledge of caregiving.

Third, I would continue to document and get more examples of his hostility and bullying before taking any action, however:

Fourth, I would turn the tables on your sister and husband, and ask them specifically how they would approach a more congenial atmosphere between the three of you.   Ask him to document the alleged "thefts" from your mother, alleged mistreatment of your sister, and any other complaints he has.  Keep them and be prepared to use them against him if he becomes more belligerent.

Fifth, ask your sister specifically how she would plan to pay for AL placement.  Does your mother have financial resources?   Perhaps sister and husband are after the money?

Sixth, has your mother executed any powers of attorney, legal and/or medical?   If so, who are the proxies?  

This situation has been raised before:  one sibling cares for a parent, another contributes nominally but either he or she challenges the caregiving parent, with threats, backstabbing, dragging in the family, etc.  It's a challenge to balance caregiving with hostel and uncooperative family members.

Some of these other threads might help:

https://www.agingcare.com/search?term=interfering%2C+hostile+family+members