、。
Without even asking me, she walked off with my umbrella.
Literal
She [topic-は] me [to-に] ask-[even-も]-doing-not, my umbrella [object-を] carried-and-went.
Two related grammar patterns layered. (1) たずねもしない is the negation of たずねる ('ask') with the intensifier も — instead of plain たずねない, the verb is split (masu-stem たずね + も + する) and the negative falls on する. The result: 'doesn't even do the act of asking.' (2) 持っていく ('carry and go away,' the te-form chain 持って + 行く) means 'take with one' — taking something to a different location. Combined with the small possessive 私の ('my'), the whole sentence captures a gentle but pointed grievance.