It is perfectly valid to reparametrize your model before implementing MCMC. Two caveats, as mentioned in the comments: (1) you need to calculate the Jacobian of the change of the variable; (2) depending on the problem, this can make it more difficult to think about the correlation between the parameters. Depending on the problem, it can be easier or harder to find a proposal that allows the chain to mix well.

However, since you are thinking of updating the components independently, another option is to forgo the $t$ distribution for certain parameters. In particular, your transition kernel can update only one parameter at each step, instead of updating all parameters at once. You can then choose a proposition which won't (at all/too often) propose values outside of the interval of support.

It is perfectly valid to reparametrize your model before implementing MCMC. Two caveats, as mentioned in the comments: (1) you need to calculate the Jacobian of the change of the variable; (2) depending on the problem, this can make it more difficult to think about the correlation between the parameters. Depending on the problem, it can be easier or harder to find a proposal that allows the chain to mix well.

However, since you are thinking of updating the components independently, another option is to forgo the $t$ distribution for certain parameters. In particular, your transition kernel can update only one parameter at each step, instead of updating all parameters at once. You can then choose a proposition which won't (at all/too often) propose values outside of the interval of support.

Edited to add: this answer by jbowman is very relevant and more detailed.