Repeating words in cases where other methods exist to express your meaning usually indicates that you are not aware of those other methods; it gives away the fact that this is not your first language.
So, while it's not wrong, there is a more elegant way to express it through the use of the word "both":
On our website, you can both buy stuff online and request delivery anywhere in the US.
If you've noticed, this takes care not only of the second "can" but also of the word "also".
Incidentally, it also turns a (halfways) compound sentence into a non-compound one, because there is really no need for a second "main" verb. The sentence becomes simpler and more streamlined (which is always preferred).
Above, I've said that your sentence is "halfways" compound because there is only one subject, which is also a bit unnatural for English. English leans more toward full "subject+verb" sets. With those parts of the sentence connected by "and", there is really no need for repetition.