How can I make frozen yogurt as quickly as possible(without using ice cream mixer)?!

I don't know if this is still relevant to you.

The fastest way would be to simply add as much sugar to the yogurt as you like until it reaches the desired sweetness. If you would like it to be vanilla add vanilla extract-- roughly 1 tsp / 2 Cups of yogurt. You could also/instead add some lemon zest-- roughly 1 tsp / 3 Cups of yogurt.

The next part depends upon what equipment you have available to you. The goal is to freeze the yogurt and remove from the freezer ever 1/2 hour to stir/blend as this helps create a nice texture. You stop continuing with this process once the frozen yogurt is so stiff when you remove it from the freezer that it's very difficult to stir/blend. You can use an electric egg beater, food processor, or just a sturdy wooden spoon to do the mixing. I would recommend a food processor.

Since you have done this without using an ice cream maker the frozen yogurt will be best if eaten on the same day that it's made. Start making it several hours (at least 5/6) before you intend to serve.

I hope that can help!