My company just acquired another company and I need to make their SharePoint site available via a different URL. The SharePoint sites are hosted by a third-party company, and I have tried their method of copying the contents (lists and documents) via a WebDAVV connection, but the version history is not maintained. Is there any way of duplicating a site and preserving document history when I do not have admin access to the server hosting the SharePoint sites?
Generally you will want Site Collection Administrator rights on the source and target site. From there, you can use a variety of content migration tools, such as Sharegate, Metavis, and others. These will allow you to not only preserve version history, but metadata and permissions as well.
3rd party tools are the option with Site collection admin rights.
But if they give you the site collection backup( .bak) file then you can restore it in your enviroment using the restore method. Only catch in this case is both Source and Destination farm should be on same Patch level otherwise it will not work.
Or you can ask them for Content Database backup.