Перевод - Китайский упрощенный -Английский - 我们希望贵学院可以派代表来参加这项比赛

Small points:

I don't think "School" needs a capital here?

Also, should it be "institute" or "college"? "School" seems like it's for younger kids? (Is Chinese different from Japanese when they say 学院 ?)

Only one "match"? 比赛 seems to refer to a larger "competition," including many matches? Again, is Japanese different?

You are right.

Better with college and matches.

But 项 directly mean one item, so it should have been one match.

学院 can be school, college, institute.

Thanks, pluiepoco!
Two final questions:

When you read the word 学院, what age do you imagine the students would be?

And I still didn't understand about the matches:

Does the text indicate that there will probably be only one match, (i.e. one game only), or will it probably be several matches, i.e. a competition?

Sorry to bother you again!

Big students not pupils,

to me,项 means one item,
not the same as 届/次, which can be many items of one year.

So could we say:

We are eager to have the delegates from your college take part in this competition.
?
(A competition contains many games)

pluiepoco?
Do you accept the version in my message above?

Yes of course. You are great!

Oh! Thank you!

I hope the weather is fine today over there in China too!

Yes, it is sunny today! Thanks God You!

God,
You want it sunny here,
then it is sunny here.