The following information may or may not be of use to you. It all depends on what kind of tourney you want to have. I only know of one way. I know there are score cards out there for euchre tournaments but I haven't been able to find them. If you have score cards or instructions that you would share with the users of this site (or know where there are some) we would appreciate it immensely!
You can use this method regardless of how many players you have. At one of our tournaments we had 24 people (6 tables). We play for 3 hours. Everybody sits down at the tables randomly when play is about to begin. We don't have teams for several reasons. First, it's nice to get to play with everybody and make new friends. Plus it does away with that team that nobody can beat! You also have to win by relying on your skill alone. If you score a lot of points with an awesome player, chances are so will someone else so it keeps things even. If you want to win, you must be aggressive and play your cards correctly. :) Second, we don't have any tournament grids to use! (Hint, hint)
A person shuffles the cards and deals one card to each player until a jack is turned up. The person receiving the jack deals. You could do this another way if you desired to do so. Play begins with that person dealing. You do not play a game to 10 points but until everybody at the table deals two times. In other words, you play 8 hands at each table and play stops after each person deals twice. Each person has a score card and writes his/her total points from that table. (Yes, it is the honor system). If you and your partner go alone three times and makes the three loaners and you call trump and make your point another three times at that table, your total score would be 15 points. When you are done at that table, it is time to move. The winners become the "movers" and the losers stay at that table. When you move, you do not play with the same person you just played with. One of the losers staying at a table will move to another seat. This way, you play with somebody different every time. When you have 16 or fewer players and only play for 3 hours, chances are good that you will end up playing with the same person again. When you move to a table to play another game and everybody has already played with each other, that's ok! Amongst friends it's easy to figure out who plays with who.
We give away 1st, 2nd, 3rd and sometimes 4th place prizes (cash) and the player with the least amount of points gets his entry fee back. We charge $5 per person. We also have a "duck" that we use which is just a little wooden figure of a duck with a clothes pin for a bill. (Use your imagination and use anything!) There is $10 put into the duck by the host before the start of the tournament. When a player goes alone and takes all 5 tricks and scores 4 points, that person gets possession of the duck (we hear "gimmie the duck" being yelled all night long). When someone goes alone and gets euchred, that person has to "feed the duck" one dollar. Usually, there's about $20 in the duck at the end of the night. Of course, the person with possession of the duck when play stops at the end of the night gets to keep the money in it.
One more note. A lot of the time, there is always that one person who doesn't show up and never lets anybody know before hand to find a replacement. What we do in that situation is use a "ghost" player. The ghost moves when he wins and stays at the table when he loses just like everybody else. The ghost is dealt cards just like an ordinary player however, the cards are placed in a pile at the empty seat. A player at one side or the other of the ghost plays the top card off of the pile. The ghost does not have to follow suit. The last time I had to play against the ghost, I went alone with 4 trump and an off ace. I got euchred by the ghost and it cost me one dollar. No, it's not the most fun playing at a table with three people but remember unless you have less that 24 people or unless you play longer than 3 hours, you probably won't have to play with or against the ghost more than one time so it's not all that bad.
We also play "Euchre Bingo" (for lack of better terms). When everybody sits down at a table, they get a piece of paper (printer paper) consisting of 5 rows and 5 columns on it (25 squares). The individual boxes contain Euchre sayings or phrases. Each paper is different. We have a coffee can with each saying or phrase in it as well. The host of the party will pull a phrase out of the can after every other round (ir whenever) and call it. The players cross the phrase off of their paper. The first one to scratch off 5 phrases in a row gets the door prize, usually a new deck of cards, a $20 bill, a six-pack, a jar of peanuts, what ever you want to use. Just make it FUN!