I'm using Vegas 10 and Panansonic 150 .mts files. I have a client that wants the final product to be H.264 with a .mov extension. I don't see any render option in Vegas for H.264 with a .mov extension.

How would I go about converting - what program should I use?

Odds are that you can get away with rendering to MOV right in Vegas by using either the Photo JPEG or Motion JPEG options.
I've never had to do it but this topic has been discussed here a lot so I'm sure that a quick search will show you several options.

You need Quicktime Pro to be able to render to H.264 Quicktime.

Sorry but you dont need Quicktime Pro to render to H.264, see pic from After Effects.
You could render using anything to MP4 and just change the extension to .mov
If you want to use Vegas to do it you just choose the MP4 codec in the Quicktime drop down.
The quality of the codec from Vegas and Quicktime Pro is very ordinary compared to other options

The MPEG-4 codec (if you meant this) available in the Vegas Quicktime render and H.264 is not the same. If you don't have QuicktimePro installed, H.264 will not be avaible for Quicktime render in Vegas Pro. Vegas Pro might differ here compared to After Effects.

But you are right, just renaming the MP4 H.264 render (as Sony AVC or MainConcept AVC) seems to do the trick.

I don't think there's anything technically preventing an application from accessing QuickTime's encoding capabilities even if you don't have a pro license. It's probably a decision made by Sony.

There's a great freeware called MPEG StreamClip, that gives you all the encoding options of QT Pro, plus some nice resizing and deinterlacing options not available in QT, so if you feel the safest option is to encode with QT in .mov, you could render a lossless intermediate and then use MPEG StreamClip. If you do that, be sure to use the b-frames and 2-pass options to achieve the best quality.

If you are satisfied with using Vegas to encode to mp4, I would still use MPEG StreamClip to re-mux mp4->mov. It's very easy (and doesn't re-encode the video/audio streams) : Just open the .mp4 and choose file->save as and then choose the MOV container.
Simply changing the extension in Windows Explorer may work, but you are not actually changing the container when you do that, so a proper re-muxing is usually a better option.

EDIT : If using B-frames or a non-QuickTime encoder, it's probably a good idea to also check with your client for any compatibility issues, because QuickTime can be fussy about what it accepts.

It's a decision made by the Quicktime manufacturer same as you can't access nor encode Quicktime MPEG-2 even with QuicktimePro without having a special - extra paid - MPEG-2 license for Quicktime installed. Then you could say SCS could have bought these extra licenses and sell them to the Vegas user for extra money. This only makes limited sense imho for Quicktime on Windows. If you want or need the access you can buy it anyway.