Surely "We want to take part" should be ok too?
That's fine, but usually more common when put in a complete sentence, in my opinion. "We want to take part in the competition".
We want to participate?
Yes, that's right
What about ‘we want to partake’?
Participate means to join in an activity. Partake means to eat or drink something that is being offered. Partake is generally used in the Mass when people go to the altar to "partake of" the bread and wine. Note the "of" btw!