Some kids do use the winter time period to look at and practice with other clubs they are interested in, this is why your coach may be suspicious.
But yes, it's totally normal to seek out training your club probably does not offer. (My club never offered anything above a rec level futsal clinic taught by non-futsal coaches.) We Have tried training at 5 or 6 different clubs over the winter periods just to get technical training that our club did not, could not, or would not teach.
I would try to coordinate the training with a friend. My kid really wanter a buddy to go with, it made big difference in motivation. There are no rules prohibiting this.