There is no explicit mention of how often the Eucharist was celebrated in the Bible.
What are the earliest historical documents that make reference to how often the Eucharist was celebrated?
The earliest indication may be in the Didache, typically dated at the end of the first century. It describes the celebration of the Lord's Supper in terms of the cup and breaking bread, and then says:
But every Lord's day gather yourselves together, and break bread, and give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. (§14)
If this is a reference to the Eucharist, then we have evidence for weekly celebration less than 100 years after Christ's death.
In the middle of the 2nd century, Justin Martyr more clearly writes that the Lord's Supper is observed every week on Sunday:
And on the day called Sunday, all who live in cities or in the country gather together to one place, and the memoirs of the apostles or the writings of the prophets are read, as long as time permits; then, when the reader has ceased, the president verbally instructs, and exhorts to the imitation of these good things. Then we all rise together and pray, and, as we before said, when our prayer is ended, bread and wine and water are brought, and the president in like manner offers prayers and thanksgivings, according to his ability, and the people assent, saying Amen; and there is a distribution to each, and a participation of that over which thanks have been given, and to those who are absent a portion is sent by the deacons. (First Apology, §67; see also §65)