〜ようにも〜ない
can't, even if (I) wanted to
Verb volitional form + にも + potential negative
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ようにも〜ない says you'd like to do something, but circumstances make it impossible: 忙しくて、休もうにも休めない, "I'm so busy I can't take a break even if I want to".
Take the volitional form (休もう, 買おう, 連絡しよう), add にも, then repeat the verb in the potential negative: 買おうにも買えない, 歩こうにも歩けない. With する-verbs, the second verb is often できない.
The first half gives the obstacle: no money, no time, an injury, bad weather, a forgotten phone number.
It's a vivid way to express frustration. たくても〜ない (休みたくても休めない) means much the same and is more common in speech.
Examples
忙しくて、休もうにも休めない。
I'm so busy I can't take a break even if I want to.
お金がなくて、買おうにも買えない。
I have no money, so I couldn't buy it even if I wanted to.
電話番号を忘れて、連絡しようにもできない。
I've forgotten the number, so I can't get in touch even if I want to.
足が痛くて、歩こうにも歩けない。
My feet hurt so much that I can't walk even if I try.
雪で、出かけようにも出かけられない。
Because of the snow, I can't go out even if I want to.