Polyesters with a high glass transition temperature above 130 °C were obtained from limonene oxide (LO) or vinylcyclohexene oxide (VCHO) and phthalic anhydride (PA) in the presence of commercial salen-type complexes with different metals—Cr, Al, and Mn—as catalysts in combination with 4-(dimethylamino) pyridine (DMAP), bis-(triphenylphosphorydine) ammonium chloride (PPNCl), and bis-(triphenylphosphoranylidene)ammonium azide (PPNN3) as cocatalysts via alternating ring-opening copolymerization (ROCOP). The effects of the time of precontact between the catalyst and cocatalyst and the polymerization time on the productivity, molar mass (Mw), and glass transition temperature (Tg) were evaluated. The polyesters were characterized by a molar mass (Mw) of up to 14.0 kg/mol, a narrow dispersity (Tg) of up to 136 °C, and low (<3 mol%) polyether units. For poly(LO-alt-PA) copolymers, biodegradation tests were performed according to ISO 14851 using the respirometric biochemical oxygen demand method. Moreover, the vinyl double bond present in the poly(LO-alt-PA) copolymer chain was functionalized using three different thiols, methyl-3-mercaptopropionate, isooctyl-3-mercaptopropionate, and butyl-3-mercaptopropionate, via a click chemistry reaction. The thermal properties of poly(LO-alt-PA), poly(VCHO-alt-PA) and thiol-modified poly(LO-alt-PA) copolymers were extensively studied by DSC and TGA. Some preliminary compression molding tests were also conducted.