I've been using this for a number of games for a while now, and figured I would share since there doesn't seem to be a program that does this for New Vegas. The non-compiled version requires Autohotkey to be installed, which can be downloaded at http://www.autohotkey.com/download/. I've only tested this on my computer, so no guarantees on how well it will work on yours.

As of 11/27/10, there is an alternate compiled version that doesn't use hotkeys!(see instructions for new compiled version below)

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Instructions for compiled version:

This way is recommended.

-First of all, start the F:NV launcher, go to options, set your resolution to match your desktop resolution, and make sure the Windowed box is checked.
-Now download the compiled version from the file section.
-Open it. (should show up as a green H icon in your system tray)
-Now, start the game, get to the main menu and press Control+LeftShift+Tilde (Tilde is the button above Tab on my keyboard).
-The game should now look fullscreen.
-If your windows taskbar remains visible, alt-tab to another window (desktop, internet browser, file, etc) and then alt-tab back to fallout new vegas. The taskbar will be gone. You only need to perform this action once.

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Instructions for alternate compiled version:

This version was written in c++ and uses a 10 second timer instead of a hot-key.

-First of all, start the F:NV launcher, go to options, set your resolution to match your desktop resolution, and make sure the Windowed box is checked.
-Now download the alternate compiled version from the file section.
-Open it, and a small box name Embiggen Window should pop up with two buttons on it. Don't hit the buttons yet.
-Now, start the game and get to the main menu.
-Alt+Tab to the Embiggen Window program and press the Embiggen button.
-Alt+Tab back into the game within 10 seconds of pressing the Embiggen button.
-Your game should be fullscreen in 10 seconds.
-The revert button will undo the "Embiggening" process.(In case you Embiggen something you didn't intend to)
-If something doesn't work, leave a comment and I'll try to figure out what went wrong.

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Instructions for non-compiled version:

This way takes a little more work, but it's more customizable.

-First of all, start the F:NV launcher, go to options, set your resolution to match your desktop resolution, and make sure the Windowed box is checked.
-Now download Autohotkey from their website mentioned above and install it.
-Once you have AutoHotKey installed download "Embiggen Window text file" under optional files.
-Open up "Embiggen Window.txt" in notepad and edit the the resolution in the fourth line( the 1280 and 1024 numbers are what you want to replace.) to match your desktop resolution.
-Now go to to the file menu and click save as. Save it with any name you want, but save it with .ahk instead of .txt as the extension.
-Now open the file you just saved (the one with the ahk extension). If a green H icon shows up in the taskbar it means it should be ready to use.
-Now, start the game, get to the main menu and press Control+LeftShift+Tilde. The game should now look fullscreen.
-If your windows taskbar remains visible, alt-tab to another window (desktop, internet browser, file, etc) and then alt-tab back to fallout new vegas. The taskbar will be gone. You only need to perform this action once.
-If you can't get it to work, it means one of us made a mistake somewhere in the previous steps. (If it was me. make sure to leave a comment and I'll correct it!)

*Thanks to jmcgrath for helping improve the instructions.

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Notes:
-You can also use Autohotkey to convert the ahk script into an exe file if you don't want to keep Autohotkey installed. (Start Menu->All Programs->AutoHotkey->Convert .ahk to .exe)
-This will also work on most other games, and almost any window. So don't hit your hotkeys when you have Embiggen Window running unless you want your current window fullscreen!
-www.virustotal.com is a good site to go to if you want to scan any of these files.


/*New in update 1:
-The compiled version should work for all desktop resolutions now.
-Most of the false-positives from virus scanners have been fixed. (McAfee-GW-Edition still refuses to cooperate though) */

/*New in update 2:
-New version rewritten in c++ is up.(alternate compiled version) */

/*New in update 3:
-Fixed small issue were the alternate compiled version wasn't reverting the window to the correct position. */