I have a bunch of scrolls handy of all different levels. My DM says I can only have access to one scroll at a time that I must choose. Does this sound correct?
For you, one
Your DM has decided the answer to this question in her game, other DMs may decide differently in different games. I can find nothing in either the rules on magic items or the rules of combat to suggest your DM's ruling is contrary to them.
Personally, I can imagine you with a bandolier of scrolls just like the rogue has a bandolier of daggers and you whipping out the one you need with practised ease. But I'm not your DM.
There is precedent for requiring an action to get an item
There is no specific RAW that I could find that addresses this question, so it's up to your DM to decide. Based on similar rules, it's reasonable for your DM to decide that you can only easily access one scroll at a time.
Specifically, the extraplanar storage items require you to use an action to get an item from them. Heward's Handy Haversack states (DMG 174),
Retrieving an item from the haversack requires you to use an action. When you reach into the haversack for a specific item, the item is always magically on top.
And the Bag of Holding says (DMG 153)
Retrieving an item from the bag requires an action.
Note that even though the item is always magically on top for the haversack, it still costs you an action to pull it out.
Finally, in the section on item interactions, the PHB gives the DM latitude for making object interactions more difficult (PHB 190):
The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle.