Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that curve that is equal to the arc length of the curve? Ex:
Have a parametric curve $v(a)$ of dimension $n$ such that $$a = \int_{0}^{a}\sqrt{\sum_{i=1}^n \frac{dv^i}{da}^2}da$$
Is this even possible besides having a linear path?
Yes. Every piecewise-smooth curve can be re-parametrized by length. Differentiate your equation and you will see that it is equivalent to the condition that $\|v'\| = 1$ for all $t$. A well-known example is the circle $(\cos t, \sin t)$.