Looking up provide in the dictionary, we see that it can be used as a transitive verb, and also as a phrasal verb with the conjunction with:
So, any of these would be valid ways to use provide:
A new mouse pad is provided to all new employees.
All new employees are provided with a new mouse pad.
The equipment custodian provides new mouse pads to all new employees.
What's the difference between these? They all essentially convey the same meaning (although the last one provides some extra details about who hands out the mouse pads), so it's largely a stylistic preference on which gets used.
Personally, as I evaluate the first two, I find nothing grammatically wrong with them, but I still think they sound just a little awkward, as if maybe a different verb might sound more natural:
A new mouse pad is issued to all new employees.
All new employees are given a new mouse pad.
Getting back to your original question, and keeping an eye on the dictionary definition I provided, I prefer the using the word with in your context:
Can you provide me with a mouse pad?
It fits the (provide someone with) construct in the dictionary, and sounds more natural, at least to me. Others might disagree. Either way, the difference is minor; you could still say the first one, and I wouldn't be thrown.