Why not use a four bit binary code on each card? That would give you 15 distinct card codes + no card present.
The cheapest minimally intrusive arrangement would be up to four small neodymium magnets epoxied to the back of each card, well spaced out, and preferably in milled or drilled recesses so they don't get knocked off, with either hall sensors, or if the scoreboard is thin enough or can have recesses routed in the back of it, reed switches. If you are using reed switches, and incorporate a diode in series with each, you can simply wire the scoreboard as a 5x8 key matrix. nd robust connectors so it can survive minimally qualified maintenance techs.
Other options would be pulsed retroreflective IR sensors looking through small holes in the scoreboard, with matte black paint to the back of the cards and squares of aluminum tape where you want the other bit state, or if drilling the scoreboard isn't an option, non-ferrous metal proximity sensors, again with aluminum tape on the cards for one of the bit states.
The only fly in the ointment for the reed switch matrix is it would make the whole project too simple - just use off-the-shelf security reed switch sensors screwed to the back of the scoreboard, using a spare terminal to fit the diode inside each, and wire the matrix to an ESP32 board or an Arduino + Ethernet shield depending on the desired connectivity, and one competent programmer could knock up the code to post to the server over a long weekend! OTOH if you've got a good hardware designer, embedded programmer and web programmer on your team you could still get enough project work out of it by designing a custom PCB, and having it manufactured and assembled e.g. by JLCPCB so it doesn't look 'arduinoish', and also to allow you to add proper input protection, and robust connectors.
The other sensor options would benefit from a sensor board for each digit so you don't have to wire up the sensors individually. Maybe give each sensor board local intelligence and use MODBUS to connect them back to the main controller board to permit simple 'daisy-chain'
Another option that may not get you much credit unless your course is heavily computer science oriented, would be a Raspberry Pi or similar with its camera module, + custom optics to get a decent full frame image of the score board, and run image recognition to 'OCR' the score board on the Pi, and also the webserver for it. OTOH it would be the cheapest/easiest for rinks to actually install - mount and cable one box where it can see the scoreboard, manually adjust aim, zoom and focus, then train it to OCR the existing cards.