Being a hoarder of monstrous proportions, is there a place to store the stuff I am collecting in Helgen Keep before I own a house?

Anise's Cabin is safe I believe.

Two part answers

Short term storage for the helgen dugeon when you start

-you can put everything on the chest in the first room of the keep
-It will be there if you return to helgen early
-you 'may' have time to go to riverwood and make it back to helgen before bandits show up

Long term early game storage
- honestly, anise cabin is distastefull, because it makes you kill a nice old lady

-instead, head up to whiterun, beat uthgerd in a fistfight
-you'll need to send back your current companion for a whaile and take utherd as a companion for 2 seconds
-grab her key ... now you got access to her house, she's never there

-on pc use console:setownership on every container for safety
(otherwise expensive items you put inside are tagged as stolen, not sure why the designers didn't thought of that)

other long term early game solution
-get a mammoth tusk for ysolda (the merchant in whiterun who want to trade with the khajits)
-get a mara amulet from riften
-marry ysolda
-you now have her house

you could also marry the girl in shore stone to get her house

faendal house also work the same as uthgerd, but it doesn't have a lot of containers

Now that you mentioned Ysolda. There is a really small chest on the side of her house next to some barrels.

The chest is safe.