We are moving to an office with a server closet that may not have sufficient depth to have a standard server rack. I found a vertical rack mount online (that mounts to the wall) that is 4U. Are there negative effects to mounting servers vertically instead of horizontally?
Certain specific case designs may have issues with mounting in other-than-horizontal attitudes, but there isn't anything inherent to server cases that would suggest this is bad. Bad case-designs would have parts vibrating loose after long periods without gravity to retain them, but this shouldn't be a problem with a major server vendor case. Most blade-servers are vertical in my experience!
There can be, but not usually - The thing is heat rises, so it is sometimes more difficult to suck the air in through the top then out through the bottom and you end up sucking in more hot air once it rises. Normally it won't be an issue - unless you have a really hot server. Then you have to make sure you have good air flow and cooling.
You can find some good wall mounted rack here: http://www.racksolutions.com/server-racks/wall-mount-racks
Or you can go with an open rack, if you have the space since putting it in a closet makes the cabinet unnecessary - Here: http://www.racksolutions.com/server-racks/open-frame-racks
It will try to suck cool air in from the front, and blow hot air out at the back. Since hot air rises I recommend mounting it with it's front facing the floor.
I've seen people mount servers this way (e.g. the folks at http://www.grcooling.com/ ) and they tell me there are no drawbacks to this, other then possible inconvenience when having to replace a component.
The short answer to the question (though already given by others) is "Yes, it is possible without any big problems". Some big server OEMs did actually provide a stand for a vertical configuration on a table.
The majority of cases I have been dealing with has a reinforced bottom to better support the weight of the server (or other equipment on top since some people like to save some rack rails and just stack a server onto another - not recommended).
But this is only done as the bottom part of the case is a big area and would bend if it was not reinforced. Since your deploying vertically, the weight will not rest on the bottom but on the side, which is also often reinforced (to support the rack rails). So no problem at all.
It also will not affect spinning parts (HDDs) or fans, if their turning axis has another direction. Compare e.g. some big storage shelfs or blade servers, where the SFF-trays are vertically mounted to a "normal" configuration with the caddies lying horizontally. If you do not move the server while operating (gyroscopic effect will then try to keep the spinning part in its original position, while the outter shell moves, which would apply bending force on the turning shaft or other damage), you can mount it in any direction.
The only real issue (as stated by @user164241 and others) could be heat - but only if you do not take the usually high wind speeds inside a server into account. Despite that, try not to counteract the hot air moving upwards (so do not mount it that the front is facing upwards).
The most logical way with regard to thermals would be @Jens Timmerman's suggestion (front facing downwards), while this is not so cool when removing storage caddies (gravity still applies to vertically mounted servers with front pointing straight down :) ), because you risk dropping them. I would personally go for a "front-facing-left-or-right"-setup.