First of all, be aware that this may not work for you. Do it at your own risk.
There are several potential ways to fix the game crashing at startup. You may not need to use all of these potential fixes:

Before anything, check your game's cache integrity (right-click on it in your Steam library).

Make sure the game is installed in the same folder as your Steam application.
This fix also works for Rainbow Six: Vegas 2.
For example, if your game is installed in C:\Rainbow Six 2\ but Steam is installed in your F:\ drive, just cut and paste the game's folder and paste it to F:\Steam\SteamApps\common\ .
When you try to launch the game via Steam, it may say that it cannot find the files. In your library, right-click on the game and "uninstall" it. It will not actually delete anything. Then, "install" it again on your F:\ drive, or wherever Steam is installed. It will not download the entire game as the files are already there.

Try to run the game and see if it works.

If it does not work, you may also want to do the following:
go to (...)\Rainbow Six Vegas\Binaries , and find this file: runme.dat.
Open "runme.dat" in a text editor like Notepad++ and replace R6Vegas_Launcher.exe by R6Vegas_Game.exe.
Warning: If you check your game's cache after this, it potentially will reverse it back to default.
Again, try to run the game and see if it works.

If it still does not work, try running the game with admin privileges (right-click on the game's exe, "run as admin") or try to run Steam as admin. It may work for some of you.

If all of this does not work, try the following fix by ChinMediaHD:

1. Open up program files from my documents and then in program files find steam. Steam > Steam Apps > Common > Rainbow Six Vegas > Direct X then scroll down till you find DXSETUP. Run DXSETUP and make sure you install it.

2. Once you finished the installation go to steam application > Library > Games > Right-Click Rainbow Six > Properties > Local Files > VERIFY INTEGRITY OF GAME CACHE. Please wait while it finishes the update.

3. Open up program files from my documents and then in program files find steam. Steam > Steam Apps > Common > Rainbow Six Vegas > Binaries. You'll need to delete the launcher and change the name of R6Vegas_Game to R6Vegas_Launcher.

4. finally you will also need to check the windows updates. Go to Control Panel > search Windows Update > Windows Up date then install all those updates if you have installed everything then you maybe fine but it might ask you to restart. if it tells you to restart then please restart because it will mean that they are fully functional.

Thanks to Baertel for this fix.
In your Steam library, right-click on your game, click Properties, then Set launch options and type "-novid" without the quotes. Confirm.