M9 nut for rear wheel axle?

From your photo, I'm guessing its a locknut, and you have a photo of the one from the other side.

The "come dig in our parts-bin" suggestion is great - any shop that helps you like this is worthy of your business. Take the whole wheel with you for testing.

Second option is to purchase a complete replacement rear axle. Admittedly its a lot more than you need or want, but spares are good. A whole axle set will likely look like this. Locknuts are labelled below.

ASIDE Given your bike is now down two locknuts, the wheel bearings will likely be quite loose. To set the bearing preload (tightness) correctly can be fiddly but is completely achievable. The only special tool needed is a cone spanner to fit your cones. If you have problems with this check back with a fresh question. (I can't find a pre-existing question on "how to adjust wheel bearings")

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As for finding/identifying a good LBS; generally speaking, bike shops fall into one of two categories.

1. Those that feel sparse or "empty" , tend to have lots of clothing, and very few bikes. These seem to focus on only selling the latest-and-greatest and won't be able to help you with a vintage bike. And they're expensive.

2. The LBS, or Local Bike Shop. May be hard to find because they're not necessarily high-street stores. Will often feel as cluttered as Grandad's shed. Also often have a whiff of oil and rubber, and will have a LOT of bikes on display... often so many that they flow outside. This is the kind of store you want.

How to find a LBS? Ask around - anyone who's a committed cyclist in your area will likely have some advise. Neighbours and coworkers are good examples.

It's not M9. M9 is never used on rears. When 3/8" axle hardware sizes are expressed in metric, they're called 9.5mm.

What you have is almost certainly 3/8"x26tpi. If you don't have a caliper and thread pitch gauge, carefully taking the measurements from the axle with a ruler works too. There is some nonzero chance it's M10x1, but not likely.

A single 3/8"x26tpi locknut is something probably easiest attained by pulling off a dead wheel or hub. It's one of and maybe the most common sizes in the world. If you're squirreling through random axle hardware, distinguishing between 3/8x24 if all you're measuring is the nut with a thread pitch gauge is impractical, so again, measure the axle.