We present the first statistical observational study detecting filaments in the immediate surroundings of galaxies, i.e. the local web of galaxies. Simulations predict that cold gas, the fuel for star formation, is channeled through filamentary structures into galaxies. Yet, direct observational evidence for this process has been limited by the challenge of mapping the cosmic web at small scales. Using miniJPAS spectro-photometric data combined with spectroscopic DESI redshifts when available, we constructed a high-density observational galaxy sample spanning 0.2 < z < 0.8. Local filaments are detected within a 3 Mpc physical radius of each galaxy with stellar mass M★ > 1010 M⊙ using all nearby galaxies as tracers, combined with a probabilistic adaptation of the DisPerSE algorithm designed to overcome limitations due to photometric redshift uncertainties. Our methodology was tested and validated using mock catalogues built with random forest models applied to a simulated lightcone. Besides recovering the expected increase in galaxy connectivity (defined as the number of filaments attached to a galaxy) with stellar mass, we show that our connectivity measurements agree with 3D reference estimates from the mock galaxies. Thanks to these filament reconstructions, we explore the relation between small-scale connectivity and galaxy star formation rate, finding a mild positive trend which needs to be confirmed by follow-up studies with larger sample sizes. We propose galaxy connectivity to local filaments as a powerful and physically motivated metric of environment, offering new insights into the role of cosmic structure in galaxy evolution.