I read this clause as something like "I myself surely, and all of those whose hearts were being made indignant, (...)" However, I'm not confident this is correct, mainly because I'm not sure what the と is doing there. What is its function?
我こそは literally means "I am the...", but this is actually an idiomatic phrase that means something like "I am the right person (to do it)" or "let me do it".
It is usually followed by と, which is a quotative particle you are probably familiar with. In case you did not know why this と is not followed by 思う/考える/etc, see だろうと how can I translate it?. So 我こそはと means "as if saying 'I am the one'", or simply "proactively", "willingly".
心驕り【こころおごり】 is a fairly uncommon suru-verb that means "to be conceited", "to fancy oneself", etc.
This いられる has an honorific meaning, and 方々 is also an honorific word.
Put together, 我こそはと心驕りしていられる方々 probably means something sarcastic like "(distinguished) people who are fancying themselves believing they can do it" or "those big-headed people who are willing to volunteer". (But I am only 90% sure about this... please do not truncate a sentence and hide the context.)