Short answer: Generally, you don't need to do softmax if you don't need probabilities. And using raw logits leads to more numerically stable code.
Long answer: First of all, the inputs of the softmax layer are called logits.
During evaluation, if you are only interested in the highest-probability class, then you can do argmax(vec) on the logits. If you want probability distribution over classes, then you'll need to exponentiate and normalize to 1 - that's what softmax does.
During training, you'd need to have a loss function to optimize. Your training data contains true classes, so you have your target probability distribution $p_i$, which is 1 at your true class and 0 at all other classes. You train the network to produce a probability distribution $q_i$ as an output. It should be as close to the target distribution $p_i$ as possible. The "distance" measure between two probability distribution is called cross-entropy:
$$ H = - \sum p_i \log q_i $$
As you can see, you only need logs of the output probabilities - so the logits will suffice to compute the loss. For example, the keras standard CategoricalCrossentropy loss can be configured to compute it from_logits and it mentions that:
Using from_logits=True is more numerically stable.