Is this proper grammar?

This is a rather charming memory of childhood.

Do you like this edit?

When I was little I was super jealous of my three older siblings because, when we went for walks with my parents, they all popped wheelies on their bicycles while I struggled to catch up on my trike.

It is a run-on sentence. You need a period after "siblings" or "because" instead of "the reason being.
Furthermore, the way you have it written, they popped wheelies when your mom suggested the walk, not when you were actually ON the walk.

1 Despite what some may think, to me saying "the reason being that" instead of "because" sounds stilted, not vibrant.

to me it is a run on sentence I would make it into less.

When I was little I was super jealous of my three older siblings, the reason being that, whenever my mom would say “let's go on a walk”, they all popped wheelies on their bicycles while I struggled to catch up on my trike.

It is colloquial diction and is alive and vibrant. All I've done is to insert a couple of commas. Changing anything else risks turning it into a flat, insipid and uninteresting narrative.

I would replace the work 'super' with 'extremely'

You need a period after When, jealous and friends