Here's where that "groom buys the flowers" comes from.
Gentlemen bring their escort flowers. Even more likely are they to do this when their escort is their sweetheart.
And a wedding is a way bigger event than senior prom.
So, the groom would bring a bouquet to his sweetheart on their wedding day. He would then, as a chivalrous gesture of affection, often give a smaller bouquet to those women closest to his bride (her mother, the bridesmaids). And he'd give his mother a nosegay as well.
That was it. No boutonnieres, no corsages for aunts and grandmas. No flowers for the front of the church, usually.
Nowadays, it's a toss-up whether anyone follows that tradition; brides generally want to completely control what the flowers are (they've chosen their own bouquets for decades and decades).
My personal vote is to throw that particular "rule" out and simply say, "is there anything the groom's family wants to do to contribute?"
As for who gets corsages, boutonnieres, etc.: In most weddings I've known about, there are bouquets and boutonnieres for the wedding party (bride, groom, attendants).
The parents usually receive corsages and boutonnieres. It's seen as a gesture of acknowledgment or respect. Grandparents also, usually.
Beyond that, it's kind of hit or miss. Sometimes the people who do readings, etc., will be given flowers. Sometimes not.
As for "do I make arrangements, or do I write a check?" I'm w/ K_Bear. This is your future daughter-in-law, and the mother of your grandkids. So first decide how much money you truly can spare for this, and then sit down with her to say, "How can I help with expenses? There's a tradition that the groom's side pays for flowers; I could do that if you like. Would you like me to take on part of the logistics in addition to the expense?"
The "proper" thing to do is whatever will work for your particular situation.