How can I remove water vapor from an airstream without a cold trap? I'm trying to perform vacuum distillation at home with a rotary vane vacuum pump.

In order to not ruin my vacuum pump, solvent vapor (in my case, water) should be removed from the air before it reaches the pump inlet. The only method that I've encountered for accomplishing this, is using a cold trap to chill the air below $\mathrm{-50^\circ C}$ to deposite the water. To this end, the cold trap is filled with a cryogenic liquid such as liquid nitrogen. 

Unfortunately I do not have access to liquid nitrogen, dry ice, etc. Is there any other affordable way to remove water vapor from the air before it reaches the pump?

Could I use a container filled with crushed silica beads or a liquid to absorb the water in the air? Or would using a very hydrophobic membrane work?
You won't easily be able to use some other liquid to pull the water vapor out of the air. I think the cheapest option is a hydrophobic membrane or filter that you feel comfortable using. A commercial desiccant system will be the highest performance next to cryogenic cooling but may not be cost effective (depends on the scale you are working at). You can also try salt in place of silica beads for absorption drying versus adsorption drying; I imagine the silica will be more efficient but salt will be cheaper. When using things like silica, salt, or commercial desiccants you may run into issues of too much pressure drop as the gas travels through the packed column. You will have to figure out how often the desiccant media needs to be regenerated/replaced to make sure your drying is adequate.