My experience...
I try to support a local parts store when I can, but have used online stores also
with mostly good results.
As Glen said, shop around.
One note, if you want to ensure a good experience and are unsure about an online
parts house, call them to verify that they actually have the part. Question them thoroughly before placing an order.
I have had two transactions that did not go so well.
The first was a few years ago with Appliancezone. They strung me along for two months telling me the part was back-ordered. I cancelled the order and placed it with another company and had the part within a week, but, I did call first to verify that they did indeed have the part in stock. Appliancezone did refund my money after being very firm with them.
The second was just recently. I placed an order with Central Appliance Parts for a part that I knew was obsolete but that they indicated that they could get. I kept a close watch on this one. After two weeks I called and wanted answers. The person I talked with put me on hold and called their supplier to check status. Ten minutes later he tells me that the supplier lied, its no longer available, I would be issued a refund, blah blah blah. That was in February and to this day, crickets from CAP. First of March I filed it with the credit card company I used and it was resolved promptly on my end. The credit card company never got a response from CAP.
So, my best advice is to call if you have doubts.