The compactness and robustness of the vacuum setup are the major limitations to develop transportable and space-borne optical clocks. The first step in the engineering challenge is to reduce volume and weight with respect to a stationary system. In this paper, we present the realization of a miniaturized vacuum system by building two anti-Helmholtz coils inside the vacuum magneto-optical-trap (MOT) chamber. The built-in coils are specially designed to minimize the distance between the coils, and in this way it is possible to reduce the current needed to realize a typical magnetic gradient of 40 Gs/cm required for blue MOT. When the MOT coil current is 2 A, an axial magnetic field gradient of 43 Gs/cm is obtained in the center of the MOT, which is enough for the first stage Doppler cooling. This novel design allows us to reduce size, weight and power consumption with respect to a traditional laser cooling apparatus, and simultaneously avoid complicating the water cooling equipment. Our vacuum system has a size of 60 cm×20 cm×15 cm, about 1/10 of the original system in the laboratory. In addition, the circularly polarized Zeeman slowing laser is sent to counter propagating atomic beam, and atoms at a few hundred meters per second are now routinely slowed down to velocities of tens of meters per second. As a result, about 16.4% of the atoms are actually trapped into the blue MOT. The final temperature of the blue MOT is approximately 10.6 mK, and the internal diameter is 1.5 mm by observing the expansion of the atoms from the MOT. The populations of cold atoms finally trapped in the MOT are 1.6×106 of 88Sr and 1.5×105 of 87Sr. The 1S0 → 1P1 transition used for the blue MOT is not perfectly closed due to the decay channel of the 5p1P1 → 4d1D2, and a part of atoms are stored in the 3P2 and 3P0 states. To prevent the atoms from losing, 707 and 679 nm repumping lasers are employed to recycle these atoms in the 3P1 state, and then the atoms decay to the ground state 1S0. Now the typical number of loaded atoms dramatically increases by 5 times compared with before. The slowing efficiency of Zeeman slower is also optimized for the operation with deceleration related to the parameter of magnet length, which uses more effectively available magnetic field distribution, in contrast to the usual constant deceleration mode. Our future work will focus on constructing a Zeeman slower combined with permanent magnets or an electric magnet for better tuning of the magnetic field.