I have the Chase club card.
The benefits started right after I was approved, even before I got the physical card - so once I got the approval notification (took about a week after I applied - approval wasn't instant), I was able to log into my UA account and see that I had club access, and was able to access a United club the next day with my boarding pass.

Never had any issue with benefits connecting with the boarding pass as long as you use the card to purchase your tickets. Since I'm gold now, this isn't as critical an issue.

Others have expressed skepticism that your son would be approved - I'll leave that up to you to deal with as of course we don't know you or your son's financial situation.