Some Americans may use "write us" in informal contexts, but the second one ("write to us") sounds better, especially for formal contexts, and is more grammatically sound. Use it.

Yes I agree with Brandon - native speakers might omit the 'to' in very informal situations but it is not grammatically correct and would often be inappropriate.

‘Write us’ is commonly used in US-English. In British-English this would be considered somewhat incorrect, but we would understand it. However, we would prefer: ‘write to us’ ☺️

2

As an addition to Brandon and Audrey's good answers, you would definitely use "write us" when you are going to add a direct object next - so "write us a message about that". The "us" is the "indirect object" of "write", and we normally signal that by using the "to". But when we chain write + indirect object + direct object (the letter, the email, the poem) in that order we drop the "to".