Does joining "IKEA Family" take any time?

When I signed up several years ago through the website I received a confirmatory email which included a bar code which I could print out and use immediately in store as a temporary membership card.
The plastic card arrived about a week later.

Yes you can. You can even wait until you check-out, tell the clerk at the register you want an Ikea family card, they will hand you a card and paperwork to fill in. The card can be scanned right away, you fill in the paperwork stepping aside and hand it back to the clerk. I did that.
If you are not comfortable with that, get a card and a form from a clerk at the register before you make your shopping and fill it in prior to going to the check-out. As you check out, hand the form back to the clerk and have your card scanned. The card is valid immediately.

You can inscribe on a PC in the store and you get a printed ticket with a bar code to give the cashier.