You can do it sans machine, but you are limited to small batches, and it tends to form crystals. You have two choices, bags or freezer container. Bags tends to make less crystallized product, and is a bit faster.
Bags
Get a large 1 gallon ziploc bag, lots of ice, a ~2 cups salt, a small, 1 quart bag and the mix to be turned into ice cream (or frozen yogurt). Chill the mix in the fridge beforehand. This helps the mix cool more quickly and evenly, meaning smaller crystals, meaning creamier product.
Place the cream mix in the small bag, and leave a bit of air. Make sure it is sealed well, you may want to double bag it. Put the small bag in the big bag.
Fill the big bag about half way with ice and add a few tablespoons of salt. Seal the big bag and start mushing the small bag from outside the big bag, moving it around and getting it in contact with the ice. You probably want gloves or to wrap it in a towel: it gets really cold. After the ice melts down, add more ice and salt, keep squishing until the stuff in the bag gets hard. You can toss it on the floor and squish it with your feet while watching TV or reading. Put it in the freezer for a few hours to condition.
This isn't quite as nice as a machine. To compensate for the extra crystallization, you can add more fat and sugar, since those stop large crystals from forming. With frozen yogurt, I don't think there is much you can do.
Freezer Container
This is easier, but makes lower quality product. Simply put your mix in a large container (leave at least half empty), and place in the freezer. Take it out every 10 minutes and shake vigorously for a few seconds. Repeat until it won't move anymore. Use a fork to fluff the mix.
Frozen yogurt made with full fat yogurt is very creamy and full flavored. You might find that the eggs are unnecessary.
If you do do the custard watch you temperature carefully as yogurt curdles easily when it is heated. If you have a lot of fat in the yogurt then it will be resistant to curdling otherwise you can take out some insurance by mixing in a little bit of corn starch- if that is allowed.
Honey is perfectly fine and many frozen yogurt recipes call for it.
Best Answer
I was sure that with all the questions and answers we have had about yogurt that this simple question must surely have already been asked. But I couldn't find it.
Making yogurt is simplicity itself.
The goal is to introduce heat-loving (thermophillic) bacteria to milk, keep them warm so that they munch on the lactose in the milk turning it into lactic acid. The acidity and warmth denatures the milk proteins and turns the whole mass into a gel.
The steps may vary a little depending on your equipment and the bacteria that you use.
"Yogurt makers" can be purchased that simply hold their contents at about 100F. Some people use heating pads. Some people set their oven on low. You can be creative- just don't get hotter than 55°C (130°F) or cooler than 30°C (90°F) or so.
To try it out for the first time I would recommend simply putting your yogurt in a heavy, lidded, container, wrapping in a towel and letting it sit in the turned-off oven for 5-6 hours.
Making your own yogurt is a fun activity. You get to pick what goes into it so there is no sneaky ingredients and you can make it for a fraction of the cost of commercial. Play around with different starters. Different bacteria will produce yogurt with different qualities and it is interesting to experiment.