When can you access the mech lab to customize your mech?

Do you mean in the campaign? And by customize, do you mean change the loadout? If yes and yes, then you can do it after the first mission (assuming that you have the gear that you want to swap in).

If you mean something else, then I'm sorry to trouble you.

Select the Mech you want to modify.
On it's display in the hanger, there is a REFIT button by the REPAIR button.
The REFIT button is how you customize each mech's loadout... but beware, there is a non-canon 'hardpoint' system in action! You will be limited as to what you can and cannot place in each area, and how many of each waepon system. For instance: a Shadowhawk with one LRM 5 can mount an LRM 20 in the same torso section, but not four LRM 5s; likewise, a Hunchback with six Medium Lasers can not swap them out for an AC 20, but could mount six Large Lasers.