As long as any sim trying to access it belong to the same household then yes, anyone will be able to access items out of any chest.
Household definition = any sim that is in your UI when controlling a sim on a residential home lot, right...so when you go to map view and click on a residential lot, any sim that lives in that house belong to the same 'Household'.
If you forget to empty chest before moving lot then it is possible that you will not be able to ever access those items again. I don't know if this is a bug or intended design.
There may be some variables as in if you move the entire Household (so all sims living on one lot) to another lot then as the game should consider this as the same Household then chest items should be accessible. I haven't tested all possible outcomes so if you feel so inclined feel free to post your results.