(first sentence for context)...because I experienced childhood without my father's love.
Would it work to also say 「幼児体験で育ったから」?
I'm also wondering if it is similar to 生活を送る, to lead a life.
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Yes, 体験を送る is understandable by analogy from 生活を送る, but 体験を送る is not a common collocation. This sentence would be more natural if it were 幼児時代を過ごしている, 幼児時代を送っている, 父親の愛情なしに育っている or something. (The standard verb used with 体験 is する (e.g., つらい体験をする), but 父親の愛情なき幼児体験をする doesn't make much sense anyway, because loveless-ness is not a one-time episode/event.)
You can check which noun is used with which verb using a corpus like this. 体験を送る is not listed as a set phrase. I suppose this is just a typo or author-specific misuse.
幼児体験で育つ would also be understood but it doesn’t sound any more natural than 幼児体験を送る, which is indeed uncommon. When the verb 育つ is used with the particle で, one would expect that で to indicate a place or environment where the person grows up. 体験 doesn’t quire meet this requirement. (It doesn’t quite make sense as a means, either.)