In an $S_N1$ reaction, the rate-determining step is the formation of a carbocation intermediate. The reactivity of the compound towards $S_N1$ depends entirely on the stability of the carbocation formed .

Let us evaluate the carbocations formed by the given options:

• Option A (Allyl chloride): Forms an allyl carbocation ($\text{CH}_2=\text{CH}-\overset{+}{\text{C}}\text{H}_2$), which is highly stabilized by resonance .
• Option B (Benzyl chloride): Forms a benzyl carbocation ($\text{C}_6\text{H}_5\overset{+}{\text{C}}\text{H}_2$), which is highly stabilized by resonance through the benzene ring .
• Option C (tert-Butyl chloride): Forms a tertiary ($3^\circ$) carbocation ($(\text{CH}_3)_3\overset{+}{\text{C}}$), which is stabilized by the $+I$ effect and extensive hyperconjugation (due to 9 $\alpha$-hydrogens) .
• Option D (Vinyl chloride): Forms a vinyl cation ($\text{CH}_2=\overset{+}{\text{C}}\text{H}$). This intermediate is highly unstable because the positive charge resides on a highly electronegative $sp$-hybridized carbon atom. Additionally, the $\text{C}-\text{Cl}$ bond possesses a partial double bond character due to the delocalisation of chlorine's lone pair with the $\pi$-bond, making it very difficult to break heterolytically .

Therefore, vinyl chloride does not undergo an $S_N1$ reaction.