The security of Internet of Things (IoTs) devices in recent years has created interest in developing implementations of lightweight cryptographic algorithms for such systems. Additionally, open-source hardware and field-programable gate arrays (FPGAs) are gaining traction via newly developed tools, frameworks, and HDLs. This enables new methods of creating hardware and systems faster, more simply, and more efficiently. In this paper, the implementation of a system-on-chip (SoC) based on a 32-bit RISC-V processor with lightweight cryptographic accelerator cores in FPGA and an open-source integrating framework is presented. The system consists of a 32-bit VexRiscv processor, written in SpinalHDL, and lightweight cryptographic accelerator cores for the PRINCE block cipher, the PRESENT-80 block cipher, the ChaCha stream cipher, and the SHA3-512 hash function, written in Verilog HDL and optimized for low latency with fewer clock cycles. The primary aim of this work was to develop a customized SoC platform with a register-controlled bus suitable for integrating lightweight cryptographic cores to become compact embedded systems that require encryption functionalities. Additionally, custom firmware was developed to verify the functionality of the SoC with all integrated accelerator cores, and to evaluate the speed of cryptographic processing. The proposed system was successfully implemented in a Xilinx Nexys4 DDR FPGA development board. The resources of the system in the FPGA were low with 11,830 LUTs and 9552 FFs. The proposed system can be applicable to enhancing the security of Internet of Things systems.