I think you’re conflating three issues. 1) Can you get a mortgage with 10% down? 2) Will the bank agree to sell you the house with 10% down? 3) Is it wise to start shopping for a house if you only have enough money for 10% down?
1) Yes, but it will be more expensive than if you put 20% down.
2) Yes, but, all else equal, the bank may prefer the buyer with larger down payment.
3) Probably not – houses are still going down – it’s possible that your house will drop 10% and wipe your down payment. Don’t spend your last dime on the down payment.