I'm sure like any other corporation the Household has processes and procedures and can't just open the vault on a whim. Security must be incredibly tight, and I can't imagine there are a lot of spare keys or access cards lying around. I would guess hazard that two people have the key to this vault HM and Angela Kelly and neither were around and weren't going to run back to London to facilitate Meghan and her hairdresser.
I think its been said Angela is the only one allowed to handle the jewels and with access to them all. Even if other people have access I can't see Angela letting them take out the tiara to give to Meghan simply because she has shown up unannounced as its on her head if anything were to happen. If enough notice could have been given maybe Kelly would have been happy to have someone else be on hand to give out the tiara after she herself had packed it up, checked it over etc. The issue seems to be that Harry didn't feel Kelly has being accommodating enough to find a date or make herself available on the date they wanted but seemingly Meghan and he hairdresser, despite nothing being confirmed, still turned up asking for the tiara. Well of course, if only Angela is authorised to hand over jewels and she is at Windsor then there is a problem.
The bit I would like to know is why they couldn't fix a date between Meghan and Kelly. I know some people on here think Angela may have become 'too big for her boots' but even she wouldn't be immune from getting into trouble if she was wilfully ignoring legitimate requests from Meghan (or their team) for use of the tiara (if this has been agreed) so either -
a) there had been no agreement/expectation/discussion about an extra session with the tiara for a hair styling. It seems Kate had not used the actual tiara for hers but a toy/replica one so maybe Angela wasn't expecting them to need another use of the tiara other than the wedding weekend
b) Kelly was given little notice of the extra date needed - if for example the only dates H&M offered or asked for were ones when Kelly was with HM at Windsor or elsewhere then it would be hard to fit in and could look like she is being stubborn when in fact she can hardly tell HM - sorry can't you change your plans...
c)Kelly really disliked Meghan so wilfully ignored requests from her in the expectation that HM (Harry's grandmother and the woman who had gone out of her way to welcome Meghan) would protect her from any fallout. This must be a very personal issue as under Kelly (or n her time with HM) the Queen has begun lending out much more jewels (to Kate and Sophie)
or IMO the more likely option d) Harry & Meghan are the original source, we are only hearing one side that overlooks something like the fact this was all in a tight time frame, that an extra date hadn't been discussed, it was their team dragging their feet or something that would make it all seem more understandable and is quite likely a bit of a filing on both sides (H&M and Kelly) and is really just a bit of a misunderstanding. Harry or Meghan have moaned / bragged either to Omid or friends who have then spoken to Omid, that an example of how they were mistreated is the tiara -
'we didn't get it when we wanted which was so inconvenient because the hairdresser was so fabulous he came from America and the slow Royal Household couldn't get us the tiara in time so we showed up (because thats what you do, thats what we nearly had to do when HM wouldn't see us after our stroppy demands went public you know!) and that mere employee Kelly wasn't even there to give it to us! Well what an inconvenience for us that my tiara wasn't ready, Harry was so furious he phoned the Queen to kick off.'
And that is where the narrative comes from, between their friends this is acceptable, it plays up to how hard done to H&M are and is just omitting a few facts so not lying in their eyes. Then suddenly this is being printed in a book and talked about and the public isn't (like their friends perhaps were) on H&Ms side but on HM's - how dare you speak to your grandmother like that, the tiara isn't yours but was being graciously loaned etc. So its time to say it didn't happen and give a - probably - more accurate telling but different from the one (slightly 'embellished') you were happy to tell to friends to help it make your point.