While it does say "your gear", that part of it is probably intended as flavor text, since it doesn't define what that actually means in game mechanics terms.
Even if you're going to insist that this means you can only do it to something you personally own, there's nothing in the subclass anywhere that says you can't then hand it off to someone else. If that were the intent, the rules would describe what happens when you try to do that, but they don't.
Can I inscribe rune into my equipment and give them to my allies?
No.
Under the rune carver power it says “You can use magic runes to enhance your gear.”
I bolded the your, because that’s the part that restricts you from passing it around. Compare it to an artificer, who can pass out the items:
“Whenever you finish a long rest, you can touch a non-magical object and imbue it with one of your artificer infusions, turning it into a magic item.”
Here it just says non-magical object, with no restrictions on who owns the object.
While it does say "your gear", that part of it is probably intended as flavor text, since it doesn't define what that actually means in game mechanics terms.
Even if you're going to insist that this means you can only do it to something you personally own, there's nothing in the subclass anywhere that says you can't then hand it off to someone else. If that were the intent, the rules would describe what happens when you try to do that, but they don't.
pronouns: he/she/they