The answer is that it depends on what is permitted by the owner of the rights in the work being copied and made available to learners.
The law of copyright applies to content that would be embedded in the VLE. Strictly making available to the public an infringing work (here to learners via the VLE - password protected or not) is further infringement. So if the content to be embedded in the VLE is an infringing copy, the institution is likely to be considered to be making an infringing copy available to the public.
This is “Infringement by communication to the public", section 20 of the Copyright Designs and Patents Act 1988 (CDPA) as amended.
Although embedding may or may not be copying, it is nevertheless likely to be considered communication to the public for the purposes of the legislation.
Consequently, the onus remains on the lecturer to clear the rights of any content that they make available to learners via the VLE - including content sourced from YouTube.
Of course, there is a lot of content that may be freely available to use in a VLE on YouTube (including
JISC Legal content). In order to reduce the risk of infringement, some time should be spent on determining the provenance of the content to be embedded. This approach may have the effect of both reducing the risk of infringement and demonstrating to learners the value and fairness of acknowledging the source of content accurately.