Politico has an article on this titled "Trump can’t run the Mueller playbook on New York feds" in which they talk about the Southern District of New York (SDNY) investigating Trump's businesses. Specifically on your question of a State indicting a sitting president, they write:
“I’m thoroughly convinced the SDNY will make its own evaluation. They will not say that’s a department policy,” said Jon Sale, a former SDNY and Watergate prosecutor who is close with Trump personal lawyer Rudy Giuliani. “They’re obviously looking at the president and I wouldn’t rule out that they could decide you can indict a sitting president.”
Trump's attack-Mueller playbook can’t be replicated in New York. For starters, the bounds of what SDNY is looking at don’t deal with Trump’s tenure in the White House, meaning any pushback on executive privilege grounds won’t fly. Trump’s lawyers have said they’ve resisted Mueller’s attempts to get the president to answer questions about potential obstruction of justice matters dealing with his time in the Oval Office. And they continue to signal the president’s team should be allowed to review the special counsel’s finished report to ensure it doesn’t violate the president’s rights.
They also quote the president's lawyer, Rudi Giuliani, who was himself a prosecute at the SDNY a long time ago (quote from the same Politico article):
“The president and his lawyers are upset about the professional prosecutors in the Southern District of New York going after a noncrime and the innuendo the president was involved,” Giuliani, who served as the U.S. attorney leading SDNY for more than five years during the Reagan administration, told CNN in December.
But in an interview with POLITICO on Friday, Giuliani downplayed any broader concerns that his former office posed a wider threat to the president.
“The same thing will happen as has happened over the last two years with all of these things. They’ll run them down and they’ll find out the president didn’t do anything wrong. Not a darn thing,” Giuliani said.
The Attorney General, William Barr, also weighed in on the matter (quote from the same Politico article):
Barr said SDNY’s work stands on the other side of a red line that he wouldn’t let Trump cross. Pressed by Democratic senators during his confirmation hearing last month, the soon-to-be attorney general said he’d protest the removal of SDNY’s head if he thought the president had nefarious intentions.
“I would not stand by and allow a U.S. attorney to be fired for the purpose of stopping an investigation,” Barr said.