Since NJ is your resident state, ALL your income is taxable by NJ, regardless of its source.
In addition, VT can tax you as a non-resident on any VT-sourced income. Work income is sourced where the work is actually (physically) performed. Thus you have to determine what percentage of your income was earned in Vermont. This would typically be determined by dividing the number of workdays in Vermont by the total workdays in the year, and then applying that percentage to your total income for the year. For instance, if you worked 250 days in the year, 25 of which were in VT, then 10% of your income would be VT-sourced.
This reference has detailed info on VT's filing requirements for non-residents, and on the definition of "Vermont-sourced" income:
https://tax.vermont.gov/individuals/who-should-file/nonresident
The same principles would apply to work done in NY, or in any other non-resident state with an income tax.
Your resident state will give you a credit for the taxes paid to a non-resident state on the dollars taxed by both, so you won't be double-taxed. In TurboTax, be sure to complete the non-resident return(s) before you do your home state return, so that the program can correctly calculate and apply the credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
