I just describe some thing need to buy when it not enough to use, for example: we have not enough equipment to performance testing activities, I need to submit ”purchase demand/request form” to manager. But I’m not sure that purchase demand or purchase request which one is correct in my sentences.
Common terms in US business are "purchase order" and "purchase authorization."
Technically, a "purchase order" actually initiates a purchase. But it is common to say something like "I sent in a purchase order" to mean "I sent in a purchase order form in draft for approval." It is a bit clearer to say a "I sent in a purchase request" if what is meant that what was sent in was a formal or informal request that someone else approve a purchase. But if it is clear that the sender does not have authority to make the purchase, "send in a purchase order" is extremely unlikely to be misunderstood.
I have seldom if ever heard the term "purchase demand." If you do not have authority to make the purchase, using the word "demand" is not exact and may be viewed as arrogant or even insulting.