In general no a religious cannot because they take a vow of poverty which means they may personally own nothing. On the other hand religious communities can and do. Religious communities must be self supporting therefore many have some kind of business activity that enables them to generate income. This can include making communion wafers, making incense, brewing beer, making and selling bread, writing icons, etc.
A secular (diocesan) priest does not have the same restrictions because he takes no vow of poverty. However, I would think he would need at least the tacit approval of his diocesan bishop. I am sure there is something in canon law that does place some restriction of priests being involved in business, commerce, trade, etc.