If you run into delays on an order, look into the
FTC Mail Order Rule.
In short, they must notify you within the stated timeframe (or 30 days if no time is stated) if they're unable to ship the item in time. If they can ship it later, they must send a notification including a revised shipping date (or a statement that they don't know when it will ship), a way for you to cancel the order, and if the shipping date is unknown, let you know that you can cancel the order at any time before it ships out. If the delay is less than 30 days, non-response to their notification can be considered consent to all this.
Any later delay notices must include the same details (revised shipping date, reason for delay, ability to cancel the order, etc.), but the customer must explicitly consent to the delay, or the order must be cancelled.
Merchants can be fined by the FTC for up to $16,000 per violation. It's in their best interest to keep customers updated and post reasonable shipping times. If Partzilla really says "Ships in 1 - 2 business days" and they don't notify you by the second business day that something won't be shipping out, they're in for a world of hurt.