Unfortunately you can't, but you should be able to buy most of the items off the shelf in Finland (except the original cardboard box).

This page displays the contents of this years box with the contents:

http://www.kela.fi/aitiyspakkaus

Here's an infomercial demoing the contents: http://www.youtube.com/watch?v=PVPw2_A9V5Y

Lot of the clothes are from finnish company Reima, which have an online shop:

http://www.reimashop.fi/

Seems not sadly, but it would make a great gift.

Lots of people will complain about how the state is getting involved in peoples lives etc so it would have been funny if we could buy them online from overseas too, thus creating a profitable business

The box is heavily subsidized and even Finns can't buy them directly. Some people re-sell theirs but mostly on finnish language baby websites.