Your questions assumes that a relay was required to switch the load and that it would normally be the "right" answer. However, other things you've said indicate that a relay might be inappropriate or at least overkill in the first place. If the load requires so little current that a 6134 opamp could supply it directly and you don't need isolation, then why do you think a relay is the benchmark to judge other solutions by? That would not be on my short list of solutions to that problem.
Instead of comparing solutions to each other, compare them to the specs. Unfortunately, you provided very few of those except that apparently you need a 12 V high side switch.
The big question of course is what's the current? If it's a few 100 mA or more (which one could infer from your original approach of using a relay), then the opamp by itself won't work. If it's only a few mA and within what a 6134 is specified for, then there's no problem using the opamp.
Since you apparently don't need isolation (another thing relay infers), there are many other possibilities for driving a high side switch from a 0-5V digital output. Two transistors and two resistors can do this easily. Since you have the spare opamp, you can use it to replace the first transistor and resistor. The opamp output drives the base of a PNP thru a resistor, emitter to the 12V rail, and collector to the load.