Not according to the 5e core rules. Those places don't call it the weave, but the magic comes from something somewhere which is the equivalent. 5e changed a lot, just like Eberron is part of the multiverse now.
According to KB, all magic (whether arcane or divine) in Eberron comes from an energy field countained and created by the Ring of Siberys. One might as well call it the Weave, except it's described, for arcane magic, as purely scientific interactions with it, and not controlled by a god, but that's a minor detail. The idea probably predates 5e.
KB said:So yes: In Eberron, you can make a cleric of ANYTHING. With that said, the description here makes clear what you’re doing. You may worship your shoe, but your shoe isn’t what’s granting you magic; you are bypassing the divine power sources and drawing your power straight from the Ring of Siberys, which as I mentioned above is essentially the source of all magic. Beyond this, I’d note the following…