I have tried to address this issue with my employer in the past. Since the same thing occurred the prior year. I filled out the W-4 form to address the issue. They gave me some weird reason for not withholding the taxes, I don't remember what they said. What other reasons apart from insufficient income would they not withhold the taxes?
If you earn more than $13,850 you would need to have taxes withheld as you would then have a tax liability.
If you fill out a W-4 requesting more to be withheld, unless you are not an employee and they give you a 1099-NEC because you are an independent contractor, they should be withholding taxes EVEN if you earn less than the $13,850. The IRS may guide them to withhold more, but never less.
"Q8: What should I do if an employee submits a valid Form W-4 that appears to be claiming an incorrect withholding amount?
A8: You should withhold federal income tax based on the allowances claimed on the Form W-4. But, you should advise the employee that the IRS may review withholding to ensure it is adequate, and that the IRS may direct you, as the employer, to withhold income tax for the employee at a certain rate if the review indicates the employee’s withholding is inadequate. Once this occurs the employee will not be allowed to decrease their withholding unless approved by the IRS."