I have a 1993 Toyota Corolla DX 1.8 liter four cylinder with 340,000 miles, the valves were re-seated and re-machined 150,000 miles ago. The main cylinder block was never re-machined. Main question being vibration through the steering column until about 5 miles. What may be contributing to this problem?

The vibration could be front tires out of balance, flat spotting or out of round, bad front bearings or worn front suspension. I would start by checking the tires out.

I agree, tires most likely. But it could be a slight misfire, good Denso iridium spark plugs last upward of 100,000 miles, but even they have their limits. If it is a minor vibration, and quits after 5 miles, then relax, no worries. With 340,000 it has served you well and can go even further, with proper oil changes, fluid checks and a slight vibration that goes away in 5 miles is of no concern. But if you mean a serious vibration, have motor mounts checked