Income tax only accounts for some of the taxes a state can take in. Property and sales tax are also forms states use to collect income, and they happen to be regressive in most cases.
If you look at the CFED chart [2], those nine states [1] with no income tax make up the top 9 slots where the bottom 20% pay many times more in taxes than the top 1% (as a percentage of income), from 3.8 times (Tennessee) to a whopping 6.7 times (Washington).
...states [1] with no income tax make up the top 9 slots where the bottom 20% pay many times more in taxes than the top 1%...
By itself, this statistic is meaningless. A state with a progressive tax system but lower rates of inequality will score poorly on this metric than a state with higher inequality but an identical tax system.
I.e., you might be showing Tennessee has a regressive tax system, or you might be showing Tennessee has low inequality. Without more details we cannot determine this.
We have states that have no personal income tax. Not surprisingly, these states are doing fine because they raise revenue other ways.