Analog Hilbert transformer
Yes, it is possible to implement a Hilbert transform in the analog domain, although it is difficult to get the desired phase shift over a significant bandwidth when you take component uncertainties and imperfections into account. There is some discussion of the topic in this document.

I'm not sure why you think you would use an integrator instead of a delay. The analog counterpart of a discrete-time delay is a continuous-time delay. This can be implemented in various ways, although as noted before, achieving precise delays in analog electronics is quite difficult. Some methods that I've seen before for inducing delays include introducing capacitive delays (e.g. with an RC filter) or using surface acoustic wave (SAW) devices.