Will taking a vacation hurt your academic career/recommendation letter?
The question has several possible answers. The right one depends e.g. on the country, the region, the university, the field, as well as the research group and the 'culture' promoted there. The answers you can expect here will therefore be mostly based on specific examples and cannot be regarded as universal.

I for instance want my PhD students to take vacations if they feel it is necessary, and they also simply have the right to do so. My reasoning is that they will come back with some extra energy, motivation and fresh ideas effectively increasing their research output. I do not know if this is true though. [Actually, I do not care if it is true. There are more important things than maximizing research output at all costs.]

In reality, the PhD students often do not take the amount of days off they could. In many cases this is because they are so involved in their research that they do not want a vacation. I sometimes send them away then. It could however also be due to pressure they feel from their supervisor. This pressure might be real or not, and often enough it is real. But: At least where I work most supervisors are convinced of the benefits of an occasional vacation. With these people, it would be a big mistake not to take a vacation when necessary and instead keep working until motivation is gone -- especially if that is only the case because you do not even talk to the supervisor about it because of fear of the outcome of the conversation.