The historical records for the Worshipful Company of Weavers are stored at the Guildhall Library site but managed by the London Metropolitan Archives (LMA) and you should contact the LMA with your initial enquiry.
Tel: 020 7332 3820
Address: 40 Northampton Road, London, EC1R 0HB
LMA offers a free basic enquiry service that provides information about the records held and the services offered. It also offers a paid document research service.
The following link will take you to the Visitor Information page for the LMA, and from there you can go on to read more about the Archives at the Guildhall Library and can find out how, where and when to access them:
Opening hours for the LMA site at Northampton Road and the Guildhall Library are different, and these should be checked on the website.