In this work, we studied the synthesis of biodegradable copolymers of the type poly(isobutyl vinyl ether)-co-(ε-caprolactone) (PIBVE-co-PCL) using a homogeneous mono-cyclopentadienyltitanium catalyst and methylaluminoxane (MAO) as co-catalyst. These copolymers can also be used as plasticizers for flexible poly(vinyl chloride) (PVC), improving its thermal properties. The copolymer PIBVE-co-PCL could be synthesized with a high conversion (>90%). The use of 39 wt.% of the copolymer in the formulation of PVC decreases its glass transition temperature (Tg) by -6.51 °C. By varying the copolymer composition it is possible to obtain PVC with different Tg values that could be used for different applications. A particular application where one could use this type of copolymer is in PVC formulations for the fabrication of blood bags. The toxicity of dioctyl phthalate (DOP), which is the more commonly used plasticizer for PVC, limits the use of these formulations for the mentioned purpose. The PVC plasticized with the biodegradable copolymer showed an increase in the degradation temperature, improving the thermal stability of the PVC formulation in comparison with the phthalates usually used as plasticizers.