Is it legal for a business to charge your credit card for a purchase at time of
processing your order and then not ship it out for 2-3 weeks or more?

According to the Federal Trade Commission, the answer is yes provided
that the merchandise is received by the earlier of the promised
delivery date or is shipped within 30 days of the order date.  If the
merchandise does not arrive on time but was promised by specific date
or has not been shipped within 30 days if no delivery date was
promised, you have the right to initiate a complaint with your credit
card company to receive a refund upon returning the merchandise or
canceling the order.

Sincerely,

Wonko

"Pay with a credit card.
Other payment options simply don't offer the same consumer
protections. Some credit card issuers offer extended warranties or
other advantages for credit card purchase. And if the product doesn't
arrive on time or if you aren't satisfied with it and choose to return
it, you can dispute your vendor's charges. In addition, if you have an
unauthorized charge on your credit card bill, your liability under
federal law is limited to $50."

"When you place an order, the vendor usually tells you when to expect
delivery. A Federal Trade Commission rule requires sellers to ship
items as promised, and no more than 30 days after the order date. If
the seller can't ship the goods within the stated or 30-day deadline,
the seller must notify you, give you a chance to cancel your order and
send a full refund if you've chosen to cancel. The seller also has the
option of canceling your order and refunding your money."

Source: "Holiday Shopping Online" Federal Trade Commission (November
27, 1999) http://www.ftc.gov/opa/1999/11/shoppingtips.htm

Disclaimer below applies, as well I am not a lawyer. I think the short
answer is "yes". But, that's why there is a way to cancel payment,
dispute a charge, etc. on most CC's. A reputable seller shouldn't
charge until shipped, but I've personally had delays of merchandise of
6 months, but that was a vendor problem and I wasn't really impatient
about getting the item.