Their By-laws state "Beginning with the year of adoption of these By-Laws, the annual meeting of the Association shall be held on the second Tuesday of May of each year." However, their fiscal year runs January to December. The Board wishes to change the date to an earlier date.
Can the Board adopt a new date without legally changing the By-Laws?
The short answer is yes, you will need to amend the Bylaws in order to change the date for the annual meeting.
Amending the Bylaws is generally a simple procedure. You'll present the amendment to the Board, the Board will agree to adopt the new amendment, and the amended Bylaws will be adopted by the Board. Keep in mind, there may be additional requirements included in the Bylaws relating to amendments. Most Bylaws simply require unanimous Board consent in order to amend.
It'd be a good idea to run the documents by an attorney just to make sure you're taking the necessary steps to comply with the provisions of the Bylaws. It should be a fairly painless process.