I would like to ask about any funding for PhD Students to buy lab equipments, reagents etc. Because I know a lot of scholarships and travel grants for PhD students, but no research grants. I know PIs and postdocs are eligible to apply for grants and provide the funding of experiments. But in many cases I cannot see any differences between an experienced PhD Student with good publications or a young Postdoc with less outstanding publication list. Neither of them will get a grant for postdocs but the young Postdoc has the chance to apply. I understand that it is necessary to make a difference between a postdoc and a PhD Student, but okay, let's provide less money as a grant or sg.
Yes. For example several of the fellowships listed as answers to the question, Options other than the NSF for PhD level grants?, will fund research expenses.
As far as I witnessed while doing my doctoral degree in Germany (in applied computer science), it was pretty much the norm that most doctoral candidates would write - or at least contribute to - one or two research grant applications in the course of their candidacy.
These grant applications were usually meant to secure follow-up funding for the doctoral candidate themselves (once their current funding runs out), or for other future doctoral candidates in the same institute (who could, for instance, build upon the work of the applying doctoral candidate).
Now, I'm not sure whether this is what you are looking for, as these doctoral candidates would technically not personally apply for the grant themselves, even though they collected all the input for the application and wrote it. However, unless I'm mistaken, it's also not the PI or postdoc who would personally receive e.g. money for lab equipment, but their institute or group, as a (depending on the size/scope of the grant, the sole) participant of the project.