Roger lists the many uses of He. We would have to find substitutes for most of them. Argon comes to mind, as it's about 1% of earth's atmosphere, a huge amount of gas. It is inert, so it could be used for many of He's uses, we would just need a way to separate it from the rest of the atmosphere.†
Hydrogen could also be substituted for some uses.
† Wikipedia: Argon is produced industrially by the fractional distillation of liquid air in a cryogenic air separation unit; a process that separates liquid nitrogen, which boils at 77.3 K, from argon, which boils at 87.3 K, and liquid oxygen, which boils at 90.2 K. About 700,000 tonnes of argon are produced worldwide every year.
Bottom line, we would have to adjust, perhaps causing some processes to be more expensive. But I think it could be done.