It is difficult to gauge by eye the sublight acceleration available to SW vessels seen in ANH. The countdown of the Death Star's approach to Yavin IV gives us very firm capabilities for both the battle station and the starfighters scrambled to intercept it.
The battle station has ~6g of sustained acceleration based on careful analysis of this countdown. It is clearly not designed for tactical maneuvering.
Fighters, on the other hand, have an acceleration around 85-200g. This is further clarified by the travel of the Millenium Falcon from Tatooine, which indicates sublight acceleration of around 80g. This suggests the lower end of estimates of fighter acceleration to be correct, as the Falcon also engages in the battle of Yavin.
We know that the mechanism driving TIE fighters are ion engines. From the VFX used for all ships, it would appear that all ships use ion engines for high-thrust applications. From the novelization, it is clear that repulsorlifts are sufficient to get into orbit, however.
Over the comm system of the Death Star, a voice announces that the Death Star is "Orbiting planet at maximum velocity. The moon with the Rebel base will be in range in thirty minutes." The assorted diagrams of the Death Star's approach to Yavin IV show that "in range" refers to having line-of-sight to the moon, as in the image below:
Averaging the Death Star and the moon's positions relative to Yavin, we find that the Death Star averages 1.5 radii from the center of Yavin (red and violet circle), while the moon averages 2 radii from the center (green circle). We may assume Yavin has a radius of 50,000±25,000 km.
A little quick trigonometry demonstrates that the gas giant protects the moon for only 136 degrees of arc (yellow), making the absolute maximum distance traveled by the Death Star (violet arc) outside of the 3.56 times the radius of Yavin. If the Death Star began traveling directly towards Yavin, this distance is at most half this.
Given a time of 30 minutes, this gives us a maximum average speed of 49±25 km/s. For a straight-line path, this would requires 55±27m/s2 of acceleration, or ~6g.
Another way to model this is to assume the Death Star to be traveling in a circle at uniform speed, based on the "maximum velocity" comment. For uniform circular motion, acceleration is v2/r, or up to ~28 m/s2. Yavin can be assumed to provide most of this acceleration.
X and Y Wing fighters were launched when the Death Star was between 20 and 15 minutes from attacking the Rebel base, beginning their attack run before the 7 minute mark. We may safely assume they operated at maximum sustained acceleration in approaching the Death Star, and spent 10±3 minutes approaching it.
This is a far simpler problem. The trigonometry mentioned in the above section (see diagram for summary; actual trigonometry left as an exercise for the reader) gives us the precise position from which the Death Star may fire. The distance is three times the radius of Yavin from the moon, or 150,000±75,000 km.
First, the X and Y Wings accelerate towards the Death Star until halted by impact with its magnetic field. This is a least-time intercept, and gives us 830±650 m/s2 - 85±66g.
Second, the X and Y Wings brake on their own, matching the Death Star's incoming velocity of around 99 kps. This gives us 2000±1600 m/s2, or 200±160 g.
During the novelization's detailed description of the heros' escape from Tatooine, we hear of the "baleful lemon eye of Tatooine shrinking rapidly." This implies that the Falcon is several planetary diameters out already. A page or so later, the Falcon is still "a few minutes yet" from a safe hyper limit, which could be as much as 6 planetary diameters from the surface.
We may therefore reasonably guess the Falcon to have traveled the last half of the trip in three to five minutes - the most likely interpretation of "a few" - and therefore traveled ~75,000 km in 5-10 minutes at maximum acceleration from zero. This means a net average forward acceleration of 740±250 m/s2, or 76±25 g.
1-10g Death Star
100+g Death Star