Another possibility -- if you muted all video tracks and checked the box in rendering labeled "Match project settings", then all you will see are audio rendering templates. In fact, even if you didn't check "Match project settings", all you will see are audio templates if you have muted all the video tracks.

Yes, make sure you have not muted the video track(s) which do have video and all your filter options have no checkmarks. If you still cannot see the video rendering options, try ticking the box next to "Video templates only" (click to open "More filter options" first) and see if your video options become available.

Thanks for the replies. To clarify, I used the option “Match media settings” in the New Project dialog, selecting the video I would be editing, so that should have set the parameters correctly. The video channel is not muted. When I try to render, only audio formats appear as choices when I have the “Match project settings” box ticked.
If I untick it, I finally get video options, but I’m not sure which of the 13 MainConcept AVC/AAC (*.mp4) templates to choose. That was the whole point of matching the original video’s settings. If I tick some of the additional filter options (e.g., Match project video aspect ratio), it reduces the number of available MainConcept .mp4 templates; if I tick others (e.g., Match project video frame rate), all MainConcept .mp4 options disappear.
I guess I’ll have to experiment. But something seems to be wrong here if I told the program to create the project to match an existing video’s parameters, but when I go to render, it’s telling me there are no video templates that match the project settings. I didn’t change any of the latter; I just cut some off of the original video.

IF there is a render option that closely matches your project it should have an "=" in front of it.

Yes. The problem is that no video formats presented as options while rendering have an equals sign next to them - none; nada; zip. Just audio formats.

Then for whatever reason, it is not recognizing the pattern of your video and audio. You will need to choose one of the closest matches and customize it for your render. Turn off the MATCH.. option in order to see all render options.

Just to follow up, I tried rendering using MainConcept AVC/AAC *.mp4 (Internet HD 1080p), since the original file was 1920x1080. Even though this template didn’t have a "=" mark next to it, the render looks fine (although triple the size of the original). So I’m not sure why starting by matching the original video settings eliminated all render templates that were supposed to be appropriate for my project settings, but I don’t care as long as I got a good result. Thanks for everyone’s help!

The "=" mark will appear next to templates that match your Project Settings. Be sure your Project Settings match the properties of your clips -- not just size, but also PAR, frame rate and audio properties.

Banquo was perfectly right and he didn't get any proper answer. Problem is Vegas is faulty: when a video being edited does not have a video format Vegas is able to replicate, then rendering options automatically offered upon selecting "match project settings" are restricted to audio only.

This is an obvious (and BIG) fault by Vegas: having selected “Match media settings” in the New Project dialog, the user expects being able to render the edited video keeping the same properties of the original one. All the more since Vegas seems to have no problem in importing it and dealing with it up until the moment one tries to render it !!!

By being forced to select a redering option NOT matching project settings editing times get MUCH LONGER !! Therefore I think Vegas should be amended, so as to warn the user when an orginal video has properties that Vegas won't be able to keep when the moment comes to render it in an edited version. Discovering it "at the last moment", after spending a long time editing, is VERY disappointing !!!!