What does 掛け here mean? Does it mean to hoist a bridge?
The 「掛け(る)」 here means "to build / throw (a bridge over/across a river)".
(It's usually written as 橋を[架]{か}ける.)
The へ corresponds to "over" or "across". (「川へ橋を架ける」≂「川に橋を架ける」)
And the 「願いの橋をかける」 here also has an implied, double meaning 願いをかける "make a wish".
(cf: 「光の川へかけた願いが、ほら、奇跡を起こすよ」) It could be Orihime and Hikoboshi's wish, but it could also be 僕たち's wish; it's about Tanabata Festival where you make a wish on stars, write it on paper strips and hang them on bamboo leaves (cf: 「その胸に仕舞い込んだ望みを、さあ書き留めて」「僕らは願う...今、奇跡は起こるよ」).
And are the 奇跡 crossing the bridge?
Yes, I think so. I think it's metaphorically saying 奇跡 is making the lovers' wish come true on this day, by crossing the river that lies between them.
光の川へ願いの橋を掛け奇跡は渡るよ
Does it mean:
"Build a bridge of wishes over the river of light and miracles will cross."
I would probably translate it the way you do, too.
I think the line has double meanings:
「天の川に橋を架けて、奇跡が渡る」
"We build a bridge over the Milky Way, and miracles will cross the bridge / the Milky Way."
and
「星に願いをかけて、奇跡が起こる/願いが叶う」
"We make a wish on the stars, and miracles will happen / our wish will come true".