From the excellent, but alas no longer canon, Jedi Path - A Manual for Students of the Force we learn that there are classes at the Jedi Temple devoted to the assembly, repair and maintenance of lightsabers. This would imply that the students themselves are responsible for the general upkeep of the Temple's stock of Training Lightsabers, under the supervision of their trainers.
Additionally, it notes that the Training Lightsabers are of a standard design, implying that they're constructed in bulk. Again, we can assume that the Jedi maintain a small stock of spare kyber crystals and standardised lightsaber parts to repair and replace any sabers that get scrapped.
A training saber consists of a hilt and an activation stud that, when
pressed, extends the blade. The size of the hilt and the length of the
blade are variable, based on the combatant’s age and size. Inside the
hilt is a crystal, lens, and emitter matrix to focus the blade, and a
power cell to energize it. In another class you will learn to
disassemble and reassemble this saber, but in this class you will
simply learn how to use it.
There's mention of a Lightsaber crafting facility within the Temple grounds in the excellent Complete Locations factbook.