I understand that on Earth, we mainly get our helium from natural gas deposits, which traps helium atoms generated from radioactive decay. Would this process still work on a terrestrial planet without fossil fuels like natural gas? If not, could there be another way for people on this planet to mine helium on a commercial scale?
You can have helium without hydrocarbons.
You are aware that the process that produces helium is radioactive decay; unrelated to hydrocarbons. Hydrocarbons and helium are associated because people looked for hydrocarbons and underground domes that can trap one gas can trap others too.
Recently some people went prospecting for helium, starting with considerations about where there would be rocks that could produce helium and geologic features to trap it.
Working with Helium One, an exploration company, the team realised this process is occurring now in the east African Rift Valley, which crosses nine African countries including Tanzania. ‘We found multiple places in the Tanzanian Rift where you have geothermal pools,’ says Ballentine. ‘In that water it’s bubbling gas and that gas isn’t carbon dioxide, it’s not hydrocarbons, it’s nitrogen with helium contents up to 10%, which is phenomenal.’ Commercial gas deposits, which contain a large amount of carbon dioxide and methane, only contain approximately 0.3% helium.
Pretty slick! So too in your world. Helium prospectors would find domes with accumulations of helium.
The nitrogen piece is not clear to me and I went digging. If anyone else wants to read up I found a thesis which goes into some detail: http://etheses.dur.ac.uk/12573/1/Full_Thesis_Danabalan_2017_with_corrections.pdf?DDD15+
I took away that there is a lot of nitrogen underground because there is a lot of nitrogen above ground. Geologic circumstances that trap gas trap nitrogen. Nitrogen is also found with hydrocarbons. If you have a dome that traps helium there will probably be other gases in there with it, but if your world does not have hydrocarbons then none of those.
If you want to produce Helium, you can always stock on alpha emitters radioactive materials, like Radon, Polonium or Americium: any alpha particle is nothing more than a Helium nucleus, which once captured two electrons and dissipated its kinetic energy is ready to use.
Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produced in other ways.
For safe to handle amounts of radioactive material the amount of produced Helium might be negligible, so be ready to use a generous amount of shielding around your production site.
You could extract it directly from the atmosphere.
At high cost but with good availability.
Admittedly the process would be energy inefficient, as the concentration of Helium is a very low 5 ppm by mass.
But the total reserves are huge. There is a lot of Helium in the Earth atmosphere.
In total, some 26800 million tons of the stuff.
This figure would be rather similar for any planet with an Earth like gravity and magnetic field. The Helium is not primordial but generated via alpha-decay of radioactives. The magnetic field is just needed to slow down stripping of high exospere Helium by the solar wind, as Helium really loves to go out into outer space.