Cylindrical vector beams (CVBs) with special polarization distribution have been extensively investigated due to the unique ways of interacting with matter. Although several configurations have been developed to generate CVBs, such as Q-plates and subwavelength gratings, the bandwidth of a single CVB is inherently narrow due to the phase geometry, which would limit its application for femtosecond lasers. Here, a broadband CVB mode converter based on an acoustically induced fiber grating (AIFG) and a tuning method of dispersion turning point (DTP) is demonstrated both theoretically and experimentally with the 3-dB bandwidth of 125 nm, which is more than 10 times that of conventional AIFGs. Not only can the DTP wavelength be tuned from the original 1500 nm to 1650 nm by thinning the fiber, but also the stable generation of a single broadband ${\mathrm{HE}}_{21}^{\text{odd/even}}$ mode can be controllably implemented by adjusting the polarization state of the incident light, owing to the larger beat length difference between ${\mathrm{HE}}_{21}$ and other CV modes. Additionally, the femtosecond CVBs and orbital angular momentum (OAM) modes are successfully generated and amplified by combining the broadband AIFG with a figure-9 mode-locked fiber laser. Meanwhile, it is verified by simulation that the choice of broadband CV mode and the tunability of DTP wavelength can be realized by designing ring-core fibers with different structures, which can furthermore improve the flexibility of generating high purity CVBs. This study provides a highly controllable technique for the generation of broadband CVBs and OAMs paving the way for high-capacity CVBs communication.