Can I upgrade the wooden crates to increase the storage?

No, but as you level up you will be able to craft a chest as well.

Yes you can upgrade your crates.

You are able to craft like 10 crates that has 15 slots storage capacity

After 10 crates are made, then the crates after that are only 10 slot capacity

In the bottom right corner when you open a crate, you will see an upgrade icon

Click on the icon and you will see what is needed to upgrade the crate to 15, 20 or 25 slots.

This is also a sneaky way to store items like planks and nails so you don't use slots in a crate as you can add them to the upgrade and save for later use.