So will they call/email me when it’s ready to be picked up?

Call/e-mail/contact the specific store (especially if this is your first gun purchase or first online gun purchase); policies may vary across stores. They will be happy to answer your questions (and get your business); if not, that's probably a clue that you may want to take your business elsewhere.

f I can’t pick it up for like 2-3 days after it arrives,

As long as you pick up your gun during posted business hours in the first seven days, it's not really a big deal. If it's gonna be more than a week (for whatever reason), it's courteous to give the gunstore a heads up. Some places may charge a "storage fee" if it stays in the store over 30 days; policies may vary.