I've offered to receivtemporarily store donations at my house for a school project.

What you've written is correct, but I think the main point of what you're saying is that you're just holding onto these donations for now. Receive generally implies that you're keeping whatever object you're getting, whether it's mail, a package, or a message.

This project is annually organized by a professor and carried out by students of the fifth year of high school.Each year a high school teacher organizes this project and fifth year students participate.

Usually, we are taught in school to avoid passive voice. That's why I have changed the sentence structure to subject + verb. I'm not sure I captured the essence of what you mean, though, by changing carry out to participate.

There's a place near a n underprivileged neighborhood which makes education more accessible for the children who live there.

Deprived is not wrong. I just think it sounds more polite to say underprivileged, or even under resourced.

it also gives them a place to stay in order for them to keep them away from the streets.

People have been bringing home stuff they didon't need anymore for months now.

When you say "have been" it implies that the action is still ongoing. Therefore, didn't --> don't helps with being consistent in your verb tenses.

We expect used stuff to be in good conditions (or new if someone wants to buy something, but it doesn't happen very often).

However, many people donate things in poor conditions.

You only need the singular form of condition, since there is only one condition being described (i.e. poor).

Apart fromDespite thatis, I was able to collect and fill twenty five backpacks so far.

We also have a project called "Nochebuena para todos", in which the students of every classroom have to fill two boxes with nonperishable food and gifts for underprivileged families.