Solution
It is possible to achieve these by utilizing the Coefficient Constraint of the General Polynomial Fit.vi. However, by utilizing the Coefficient Constraint input, instead of making the value a to be always positive, you can set the value a to a specific value.
Using the Regression Solver.vi sample program which can be searched using NI Example Finder in LabVIEW as an example, you can set the value a to a fixed value by setting the order to 3 and enter a positive value to coefficient. By clicking Run, you can see that the value of a at the Equation of fit will be a positive value that you had entered.

You can also utilize the Coefficient Constraint input to only fit first-order and third-order polynomial, ax3+cx. To achieve this, you need to configure the order of the Coefficient Constraint for both 0 order and 2 order to 0.
