According to the NYT, yes, he does better in online polls than phone polls.
Not exactly. He actually gets slightly more support in live phone interview polls. But his
margin of defeat to Clinton is bigger, with Clinton also getting more support in live phone interview polls. The difference is that the internet polls tend to have a larger share of voters saying “I don’t know” or “other”. Thus, the “shy Trump supporter” theory doesn’t make any sense.
This comes up every two years, whenever we have a discrepancy between live phone interview polls on the one hand and robo-polls and internet polls on the other. Someone always starts claiming that the latter polls are the accurate ones, because people are too embarrassed to tell the truth to a live human being. But there is absolutely no evidence that the robo-polls or internet polls are more accurate than the live phone polls, so I don’t know why people keep believing this stuff.