Wide-band-gap two-dimensional semiconductors have extensive applications in high-power electronics and optoelectronics in the blue to ultraviolet region. In this study, we investigate the electronic, mechanical, transport, and photoelectric properties of $\mathrm{Sc}$X$\mathrm{I}$ (X = $\mathrm{S}$, $\mathrm{Se}$, $\mathrm{Te}$) monolayers using a first-principles method. Some conceptual nanodevices based on $\mathrm{Sc}$X$\mathrm{I}$ monolayers are constructed, such as p-n-junction diodes, field-effect transistors (FETs), and phototransistors. Their multifunctional properties are subsequently revealed. The results indicate that $\mathrm{Sc}$X$\mathrm{I}$ monolayers, all of which are semiconductors with a moderate direct band gap of 2.42--1.34 eV, show many interesting properties, such as high dynamical, thermal, and mechanical stabilities, low cleavage energy, significant mechanical anisotropy, relatively low stiffness, and electronic properties that are tunable via applying strain. Additionally, the p-n-junction diodes of the $\mathrm{Sc}$X$\mathrm{I}$ monolayers display a strong rectifying effect and remarkable electrical anisotropy behavior. Moreover, the gate voltages effectively regulate the current through the FETs of the $\mathrm{Sc}$X$\mathrm{I}$ monolayers. $\mathrm{Sc}$X$\mathrm{I}$ monolayers and their phototransistors also show good photoelectric responses in the visible and ultraviolet regions. Strain can tune the device transport and photoelectric properties of the $\mathrm{ScSI}$ monolayers. Our results suggest that $\mathrm{Sc}$X$\mathrm{I}$ monolayers can be an alternative platform for flexible applications in microelectronic nanodevices, especially in photoelectric sensors.