Finding the Uncertainty in a Spectral Line
If you are doing chi-squared fitting, then a value for the uncertainty (or an estimate) should have been returned by the software. If not, you could perhaps manually estimate by fixing the line centre at various values, allowing the other parameters to be fitted, and seeing where the minimum chi-squared increases by 1 over the global minimum.

Failing that, output your spectrum as an ASCII file and fit it with something that does provide uncertainties like gnuplot.

Failing that, here's a rule of thumb. The line centre is determined with a precision given by the FWHM divided by the signal to noise ratio per pixel.

Note that none of the above take any account of uncertainties in the wavelength calibration.

EDIT: It appears that lmfit does estimate uncertainties. https://lmfit.github.io/lmfit-py/confidence.html