It's always worth contacting the dissertation adviser. Aside from that option,
dissertations are increasingly available online, typically requiring interlibrary loan services. The company involved is called ProQuest
here. Their search form returns this kind of header, followed by a lengthy abstract:

A topological construction of Vassiliev style invariants of links
Day, Colin; Stasheff, James D. The University of North Carolina at Chapel Hill, 1993. 1993. 9324015.

To get beyond this public listing, direct purchase or interlibrary loan access then has to be used (which I haven't actually done). Keep in mind that most dissertations don't have lasting scholarly value in themselves. But Colin Day left mathematics soon after his degree and didn't publish further. I hosted his visit to our department when he was considering graduate schools and am saddened to hear of his premature death.

ADDED: Maybe it's useful to quote the online abstract of the thesis (symbols edited).

"An explicit construction of Vassiliev style link invariants is developed; a new construction of Vassiliev knot invariants is shown in the process. Inside the space of all maps $\mathcal{L}_n$ from $\prod_{i=1}^{n} S^1$ to $\mathbb{R}^3$ is the discriminant $\Sigma$, which is the set of maps that are not embeddings. Following Vassiliev, we employ a spectral sequence to find certain homology groups of the discriminant and, using a sort of infinite dimensional Alexander Duality, obtain knot and link invariants which are elements of $\widetilde{H}^0(\mathcal{L}^{n} - \Sigma$) and are link invariants of finite type. To calculate the value of the invariant on a knot or link we need some initial data; the needed data is entered into a table called an actuality table. Following Vassiliev, an inductive algorithm is developed that allows us to trace the progress of a given cycle through the spectral sequence without having to calculate the entire spectra sequence. We carefully investigate the geometry of the discriminant in the course of this development.

"We give examples of how the invariants are derived and how they are evaluated on links. The examples presented include examples of knot invariants, link invariants and invariants of link homotopy; we evaluate one of the invariants of link homotopy for two component links on a whole class of two component links. We also give the generalization to links of Birman and Lin's result that establishes an important relationship between invariants of finite type and a general form of the HOMFLY polynomial."