Love this thread. We went through the exact same thing when our son was 2 3/4. He wanted an electric guitar like his Daddy's. (I was secretly hoping for Suzuki cello.) On a side note, at this point he had been playing on a drum kit since 2 and sometimes on his Mama's kit.

Super passionate about music since he could talk.
Now, electric guitars. The problem with 1/4 size electric guitars is there is no truss rod so they are very difficult to tune. We tried a couple at a local music store and as soon as my husband tuned it it went out. Through lots of reviews I found one on Amazon for around $120. If you're interested I can look it up. Also, a 1/4 size electric guitar is VERY HEAVY for a little dude, but somehow they manage!
We tried a few lessons but it's hard for little fingers! What we ended up doing was putting a stand in his room with an amp, microphone (my DH and I have gear) and his little drum kit and he could jam out whenever he wanted. Between 2 3/4 to 5 he played a lot of drums and guitar. Wrote a lot of songs. We do Family Jams every Sunday morning still. I highly recommend noise canceling headphones if you get drums. He would sit at his kit and sing super loud while jamming out! He started formal piano lessons at 5 and I think because we backed off and let him mess around we timed it perfectly. He has become very serious about piano, but is now learning chords on his guitar. It's so much easier now that he can read music!
We go to a lot of local music shows and in store performances where he gets to meet local musicians. Very empowering!
Have fun! Your adventure is just beginning,