Cut $V$ with a plane passing through $p$ and $q$ and reduce to the case of plane curves. Through $p$ there are only finitely many branches of the plane curve (and one of them contains $q$). For each of these branches there is a local parameter $t$ and the branch can be expressed as the image of a map given by power series in $t$ with $t=0$ mapping to $p$. Now, it may not start with $t$ but take the lowest power of $t$ occurring in the parametrization, say $t^r$ and replace $t$ by $t^{1/r}$. Now you end up with a Puiseux series in the new $t$ the way you want it. Standard books on plane curves (e.g. Fulton) will have proofs of the statements about branches and local parameters to fill in the details of this argument. There is also the issue that things might depend on $q$, I don't think that's a big deal.

Cut $V$ with a linear subspace of appropriate dimension passing through $p$ and $q$ and reduce to the case of curves. Through $p$ there are only finitely many branches of the curve (and one of them contains $q$). For each of these branches there is a local parameter $t$ and the branch can be expressed as the image of a map given by power series in $t$ with $t=0$ mapping to $p$. Now, it may not start with $t$ but take the lowest power of $t$ occurring in the parametrization, say $t^r$ and replace $t$ by $t^{1/r}$. Now you end up with a Puiseux series in the new $t$ the way you want it. Standard books on curves (e.g. Fulton) will have proofs of the statements about branches and local parameters to fill in the details of this argument. There is also the issue that things might depend on $q$, I don't think that's a big deal.

Cut $V$ with a plane passing through $p$ and $q$ and reduce to the case of plane curves. Through $p$ there are only finitely many branches of the plane curve (and one of them contains $q$). For each of these branches there is a local parameter $t$ and the branch can be expressed as the image of a map given by power series in $t$ with $t=0$ mapping to $p$. Now, it may not start with $t$ but take the lowest power of $t$ occurring in the parametrization, say $t^r$ and replace $t$ by $t^{1/r}$. Now you end up with a Puiseux series in the new $t$ the way you want it. Standard books on plane curves (e.g. Fulton) will have proofs of the statements about branches and local parameters to fill in the details of this argument. There is also the issue that things might depend on $q$, I don't that's a big deal.

Cut $V$ with a plane passing through $p$ and $q$ and reduce to the case of plane curves. Through $p$ there are only finitely many branches of the plane curve (and one of them contains $q$). For each of these branches there is a local parameter $t$ and the branch can be expressed as the image of a map given by power series in $t$ with $t=0$ mapping to $p$. Now, it may not start with $t$ but take the lowest power of $t$ occurring in the parametrization, say $t^r$ and replace $t$ by $t^{1/r}$. Now you end up with a Puiseux series in the new $t$ the way you want it. Standard books on plane curves (e.g. Fulton) will have proofs of the statements about branches and local parameters to fill in the details of this argument. There is also the issue that things might depend on $q$, I don't think that's a big deal.