"As I write" can mean "even as I write," which is in parallel to the stock phrase, "Even as I speak" -- something that is happening at that same moment.
"Even as we speak, the Martians are invading Tokyo!"
"Even as we speak, your ex-girlfriend is making out with your brother!"
"Even as I type this, someone is posting something else!"
Or it might be a synonym for "while I write"!
"The cat watches me as I write."
"The radio is playing as I write."
Or, if you are watching the anime Princess Tutu or reading the webcomic Namesake, it means that things are happening because you write them happening. Because in fantasy settings, Writers are dangerous that way.
"The people believe as I write [them to believe]."
"The duck dances as I write [her dancing]."
"The crow princess is eaten by her father as I write."
"The card soldier becomes friendly as I write."