To convert an existing text[] array to the enum type letter[] (like you ask) simply cast:
SELECT '{b,c,a}'::text[]::letter[];
To actually convert the table column (like you probably wanted to ask):
ALTER TABLE sample ALTER letter_data TYPE letter[] USING letter_data::letter[];
Triggers a whole-table rewrite. Only some noted exceptions allow in-place conversion. See:
We need the explicit cast in a USING clause since, quoting the manual for ALTER TABLE:
A USING clause must be provided if there is no implicit or assignment cast from old to new type.
fiddle
Aside: I rarely use enum types. There is often a better solution.