Anyone who wants to throw a shower may offer to do so. Your future mother-in-law and future sister-in-law did not step on any toes here.
In some circles it's actually considered rude for the bride's mother to throw the shower, in fact. But not all circles. It might be "traditional" for the Maid of Honor/bridesmaids to throw a shower - but, again, they need to OFFER. It's not their mandatory job. If nobody offers to throw a shower then the bride simply doesn't get one.
Your mom needs to get over it. Especially since she wasn't in a financial position to host a shower in the first place. What was her plan here ... to not throw a shower because she cannot afford it, and therefore no one else is allowed to throw you a shower? She needs a reality check here.
She can throw you her own shower if she wishes - again, since the shower is an optional gift that someone may volunteer to give, there really aren't any limits on how many a bride can have. So long as the bride is not throwing/demanding them herself, that is.
Or your mom can just show up to the one FMIL and FSIL throw with a smile on her face. And be thankful that she doesn't have to pay.
Or she can contact FMIL and FSIl, and offer to help them with their shower, either with a monetary contribution or by offering something else - setup/cleanup duties, bake or buy the cake, address the invitations, plans the games and prizes, etc. But if FMIL and FSIL say "no thanks" then your mom needs to be gracious and back off.
If your mom wants to throw a non-shower event, like a brunch or a tea, she's certainly free to do that as well. Again, though, not mandatory or expected. (Although, I have to add ... your mom throwing a brunch the day after your in-laws' shower would seem REALLY passive-aggressive, so if your mom wants to do this then I'd encourage her to space it out. And make sure the future in-laws are invited.)
Whatever happens, though, you really need to stay out of it. You should have no part in planning a party where the main objective is for people to bring you gifts. Frankly your mom is being a pouty baby here, and it's not your job to pacify her. Just do your best to stay out of it and let her handle it herself - she's a big girl.