No - if your dish is the legacy type LNB (4 outputs on the LNB) then you MUST run the 2nd coax.
Only DISH has the "splitter" thing behind the receiver to run both tuners from 1 coax - D* does NOT use this on their SWM technology.
If you have the SWM LNB (ONLY 1 output on the LNB & you have a power supply inside) then you just connect the single coax to the SAT1 input on the receiver & you're all set.
I'm guess you have the legacy dish, so if you can't/don't want to run the 2nd coax, you need to change the LNB to an SWM - you probably would be better off calling D*, as if you just got this installed, they should have never left you with only the 1 tuner running.