Yes, safeguarding of adults training is mandatory. In fact, it is a legal requirement for anyone who works with vulnerable adults and if they fail to prove that they have completed a safeguarding training course, they could face prosecution.
Organisations also need to follow safeguarding legislation such as the Care Act 2014 and The Safeguarding Vulnerable Groups Act 2006 to ensure their employees are deemed suitable to work and are following the correct procedures when it comes to vulnerable adults.