Are there any side quests that become unavailable after a certain point in Dragon Quest IX or can I come back any time and do them all (for example after finishing the main storyline)?
Nintendo shut off the wifi for this game (along with all other DS titles) As of May 2014. Because of this, you are no longer able to download any of the DLC quests, making all of these unavailable at this time. If you had them downloaded from before this time, you should still be able to complete them.
Otherwise, no. At any time (assuming you have the means to get there), you can go back to any area of the game and finish side quests there.