If the film or TV project you appeared in is somewhere online, there are online services that can get the clip for you. Actors have used ripit.me or clipgrab.org, which has a program for both Mac and Windows. Expect to pay a small fee however.
If the project is posted to YouTube, you can easily download the whole thing to your desktop by simply replacing the “www.” in the url with “ss” (without the dot). This will redirect you to Savefrom.net, where you can click on "download video in browser" (look on the lower right third of the screen) and select the video quality you prefer. Done in less than a minute and free.
If the film/TV project itself isn’t posted to YouTube, see if you can at least find the trailer, which still gives you something to post on your websites. You can also use software (I use "Grab") to capture stills from the trailer, which are in the public domain since a trailer on YouTube is already public.
The best strategy is getting a clips commitment from the producer upfront, either when you sign the contract or during shooting, and establishing a time frame, as in "you will get HD clips within three months of shooting the film." Follow up after filming with a 'thank you' email reminding him/her of the conversation; say you just want raw footage (no music or effects), and ask when would be a good time to check in on the finished product. Then follow up at that time. If the clips aren’t forthcoming, keep emailing every few weeks until you get them. Persistence is usually successful.
With students, remind them that they can email the clips for free via WeTransfer (or other such sites) and provide the link. Be sure your deadline is before they graduate from film school, and start your time frame with the end of shooting. Don't say "three months from finishing the film." I made that mistake with one student filmmaker and it's amazing how long it's taken him to "finish."
With student films and other low budget productions you can also do as one Los Angeles actor does: put it in your contract that you get useable HD clips within three (or six, you decide) months of shooting or the filmmaker agrees to pay an additional $400. This strategy reportedly has an amazing effect.
When you get into larger productions your agent may be able to help, although once you start appearing regularly at that level you won't need a reel because everyone will have seen you.
Actors live for that day.