Yes, astrophotography certainly can be done without auto guiders. It was done that way for many, many years, using the tedious and finicky process of manual guiding. You’d peer through a guide telescope attached to the main telescope, equipped with a reticle eyepiece. As your the reticle moved off your guide star, you’d manually adjust the telescope, either with a mechanical device or with an electric motor.
There are other ways you can do astrophotography without an auto guider. One method is to have a high-accuracy, high-precision mount that has been very accurately and precisely polar aligned. I had a chance to use one of these gems quite a few years ago and got a 30-odd minute unguided exposure with beautifully round stars.  That’s an expensive solution, requiring a rock-solid mount that’s been to set up and aligned with great care.
You can go through some bright targets such as the Moon, planets, brilliant multiple star systems, and even some of the more luminous deep-sky objects (DSOs).
Astrophotography Without Auto Guiders
Most astrophotography utilizes long exposures. I shoot for 300 seconds when I can. This means my mount (an iOptron iEQ45) needs to track for 300 seconds with minimal movement of the target in the field of view.
The iEQ45 isn’t bad, but it isn’t perfect. Imperfections in the gearing and timing of the motors mean it won’t correctly follow the target. But if my polar alignment is excellent and my payload is very well balanced, it does a pretty good job – such that in 300 seconds (5 minutes), there’s usually no noticeable tracking error when I’m imaging through my 80mm f/3.75 refractor.
When we talk about “tracking,” its a bit of a misnomer, tracking typically only means the way the clock-drive of an equatorial telescope moves along the RA axis in the sidereal rate. The term tracking suggests an active correcting mechanism, but this is not the case – it’s merely moving at about 15° per hour.
With an equatorial mount that is appropriately aligned, you need only move the mount in the RA axis at the same rate the earth is spinning to compensate for the earth’s rotation. If your mount is doing that correctly, any kind of active correction is unnecessary. But that level of precision is not easy to obtain, and the higher the magnification of your optical train (camera, telescope, etc…), the more noticeable any deviation from perfect will be. That same mount that can do 300 seconds with my 80mm can usually only do about 60 or so with my 8″ SCT (f/10 but reduced to f/6.3).
Time Before Computer Auto Guiding
Before computer auto-guiding, astronomers used to have to sit at an eyepiece and work to keep a guide star in their cross-hairs, making minute adjustments manually. Now, auto-guiding does this for us, making the process significantly easier on us all. But if you have a good mount, well-aligned, well balanced, and your focal length isn’t all that long, you can probably do some decent long exposures. My club (the Houston Astronomical Society) recently installed a 12″ RCOS in our observatory on a Paramount ME. I haven’t had the chance to use it, but one of the guys who’s been working on it says he can do a 10-minute exposure without guiding at its native focal length of about 2400mm. That’s not bad at all.