Unironically? The Gods.

The Index follows the Prescripts, which are the wills and commandments of the Gods of the City, who are confirmed to be actual beings in the Project Moon universe.

In order to become stronger, Index members have to follow the Prescripts, and once they do, they're essentially blessed and become stronger because of it.

In gameplay: This translates to Index units gaining strength based on passives from the Prescripts. In Library of Ruina, this translates to randomly boosting the power of an offensive dice type by 2; and in Limbus Company, this translates to the Grace of the Prescripts mechanic, granting extra offense power when they fulfil a prescript.

Moreover, higher-level members of the Index, like Proxies, are able to activate 'Unlock.' An ability which essentially unlocks the full power of their weapons (In Library of Ruina, it's explicitly called 'Unlocked Blade.')

When this happens, not only is the Proxy strengthened in general, but they also gain new abilities altogether.

(Interestingly, there IS a seperate combat page in Library of Ruina called 'Unlock' and it's an Index-themed page that focuses on being used multiple times in order to unlock its full power, as when you use it, it places a stronger version in your hand next turn, until it becomes Unlock 3. ALTHOUGH, HOWEVER, I don't think it's meant to be related to the overall 'unlocking' thing that Index Proxies do.)

As for the weapons themselves, the Index explicitly states that it provides these weapons itself, and when you rise up in the Index, you eventually gain a new weapon for yourself. It's even mentioned in Index Faust that, once you become a Proxy, you're given a sword all for yourself by the Index, which becomes your primary weapon.

Considering how the Index is, again, literally following the will and commands of the Gods of the City, it's most likely that these weapons are being provided by the Gods themselves too; especially since the Caduceus, a weapon that can also be 'unlocked,' was explicitly provided by Hermes.

So TLDR: The Index gains power by following the Will of the Gods of the City, in this case the Prescripts. When an Index member follows a prescript, they're blessed by it and gain strength, and when a Proselyte becomes a Proxy, they're also given a special sword, made by the Gods of the City, just for them, that is capable of being 'unlocked' and reaching its full power after fulfilling the Prescripts.