I know the ideal way to store fish in the freezer is to use a vacuum seal, but does anyone have any tips on what to do if you don't have one handy?
If you wrap the fish (or any meat you're freezing) with a layer of plastic wrap and a layer of aluminum foil, it will keep out air out and moisture in as much as possible without vacuum sealing.
Make sure you wrap with plastic wrap first, then aluminum foil. The taste of aluminum can leech into the food and supposedly there are health risks associated with aluminum/meat contact as well, though I haven't heavily researched them. Better safe than sorry, though.
Then I also recommend using a straw to suck out as much air as possible, as chris suggests.
This article has some great guidelines on freezing meat and minimizing freezer burn. While it only calls out meat and poultry, I use the same methods for fish with great success.
We portion our fresh fish into what we would eat at a meal, place in a ziplock bag and fill with water, squeeze out as much air as possible, zip shut and then freeze. If we have company, we take out 2 or 3 bags. Usually the fish are cleaned and filleted. This keeps the fish from freezer burn. Has worked great for us for many years.