How about this: “It was my grandma’s birthday and we wanted to make it special! My mom, my grandma, and I all went to her favorite store.” I’d argue that using “my grandma” instead of “her” would be unpleasantly repetitive there.
That approach is marginally better. It’s still arguably unclear. The store could be your mom’s favorite store to get special gifts from
If you think using Grandma three times is too much, you could replace the first Grandma in the second sentence with “she.” For instance, “she, my mom, and I went…”
My mom took my grandma and I to her favorite store
My grandma and I went to my mom's favorite store with her