Is it normal to be rejected by many firms at the application stage for vacation schemes/training contracts?

Very.

The competition for training contract and vacation scheme places is fierce. There are far more eligible candidates than places. Many strong candidates get rejected for a range of reasons.

Vacation schemes are typically harder to get onto because a candidate can do multiple schemes at different firms but can only accept one training contract.

That’s why it’s important to increase your odds and apply to more than a handful of firms. I previously answered how many firms you should probably apply to in this article.

I’d also consider whether any of your applications have red flags (described in this article) that could be hindering your chances.