Is it possible to create a cycler that can travel from Venus to Mars, then back to Venus?
Yes. When only two planets are involved, there are several infinite families of cycler orbits.
Given the 434 day long orbital period of an ellipse touching both Venus and Mars is almost twice as long as the 225 days orbital period of Venus, it's even possible to construct a cycler that has a Mars-Venus transfer leg directly after a Venus-Mars transfer leg. The extra two weeks are trivial to make up for with a slightly higher apoapsis and the bending angle of the Venus or Mars flyby.
It should also be possible to use the cycler quite often, since the ratio of the Mars-Venus synodic period and the orbital period of Venus is 1.485, making the the angular difference between every other planetary lineup very small. (within what the flybys can make up for).
Using as little delta-V as possible
The cycler itself would not need to change its velocity. Accelerating up to, and down from, the cycler should be very comparable to a regular Hohmann transfer due to the very similar shape of the cycler orbit.
Note that no cycler save any transfer delta-v
it would need to travel from Venus to Mars with a 1,000,000kg(1000 tonne) payload
Payloads are not relevant to cyclers. Cyclers are not a more efficient way of moving cargo. They are a reusable orbital "home", so you don't have to accelerate and decelerate your shelter for every trip.
Would light-sails be a practical propulsion method? And is it possible, with near-term technology(super-heavy-lift rockets, more efficient solar panels, ion engines, etc...)
Boarding/disembarking a cycler requires high thrust impulsive manoeuvrers. Thus solar sails and ion engines aren't very useful here.
For the general problem of moving cargo between Venus and Mars however, which sounds like the problem you are actually trying to solve, low-thrust propulsion is a viable alternative.
Perhaps you want fast cyclers for passengers, slow ion crafts for cargo?
An explicit cycler solution. (many others are possible);
After 4 synodic periods of Venus and Mars, the planets line up 21.3 degrees earlier.
- Transfer orbit periapsis: 0.995327 times Venus semi-major axis.
- Transfer orbit apoapsis: 1.0377 times Mars semi-major axis.
From that, we can calculate an angular distance between inbound and outbound Venus orbit crossing to be 21.3 degrees.
The orbital period (450.8 days) of this orbit is also such that if the cycler leaves Venus on the outbound crossing, it meets up again with Venus on the inbound crossing 436.1 days later.
The Mars-flyby in-between picks a high altitude, to not alter the orbit significantly.
The inbound Venus flyby places the cycler into a storage orbit exactly double the orbital period of Venus (449.4 days) (Only requires a very gentle flyby angle since we're already close to that).
After two orbits in this storage orbit (898.8 days), the new Venus flyby is now the outbound flyby, and the process repeats 21.3 degrees shifted.
Or as a timeline:
- Day 0: Leaving Venus
- Day 174: Mars flyby (Option 1, fast Venus-Mars, slow Mars-Venus
- Day 262: Mars flyby (Option 2, slow Venus-Mars, fast Mars-Venus
- Day 436: Venus Arrival
- Day 885: Optional Venus flyby
- Day 1334: Same as day 0
This pattern keeps repeating itself, since every cycle rotates this by 21.3 degrees, the same as the angular shift of the Mars-Venus synodic lineup.