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How much current can I safely pull from an ESP8266?
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5V will be pulled directly from 5V supply
The chip you are concerned about getting warm I believe is your 3.3V regulator.
I would ensure your transistor has some resistors to ensure you are not drawing more current than required from the 3.3V supply
If that is all good then I will remind you of a few things
I couldn't find an exact Datasheet but I am led to believe that the chip is a clone of this
https://www.sunrom.com/download/670.pdf
Which I found on this store page
https://www.sunrom.com/p/rt9193-33gb-rt9193-33pb-sot23-5-300ma-ldo
And here is a wiki entry incase you are curious about Voltage Regulators and the such
https://en.wikipedia.org/wiki/Low-dropout_regulator