Very cool question. I want to get into a little bit of detail here because otherwise there would be a one-paragraph answer, and I don't think that would cut it. So here goes.
The planets in the solar systems have orbits with pretty low eccentricities (see this for more eccentricity values). At the upper end is Mercury, with an eccentricity of 0.2056. At the lower end is Venus, at 0.00677. Earth is in between but moderately low, at 0.0167. The distance between perihelion and aphelion is 5 million kilometers - in an orbit with an average radius of about 150 million kilometers. Note, though, that eccentricities are always changing.
We would certainly have "seasons" if there was no axial tilt, but they most likely would not be dramatic. As Wikipedia says
Because of the increased distance at aphelion, only 93.55% of the solar radiation from the Sun falls on a given area of land as does at perihelion.
But on this page, it says
Orbital eccentricity can influence temperatures, but on Earth, this effect is small and is more than counteracted by other factors; research shows that the Earth as a whole is actually slightly warmer when farther from the sun. This is because the northern hemisphere has more land than the southern, and land warms more readily than sea.
So more land means that the planet can absorb heat better, which counteracts the change in distance.
But I think that's pretty boring. Don't you? So let's calculate how eccentric Earth's orbit would have to be for it to have seasons without axial tilt. From the chart here we can see that the highest average hemispherical temperature is 22 degrees Celsius, while the lowest is 8 degrees Celsius.
What about other planets?
There's a steady downward trend as we get further away from the Sun (Venus is an exception because of its runaway greenhouse effect). For Earth, the average surface temperature is 287 K (15 degrees C). So our maximum would be 295 K and our minimum would be about 281 K (Kelvin = Celsius + 273). Compare that to the other planets.
I made a graph, which unfortunately I cannot paste here, that shows that to have a surface temperature of 295 K, the planet would have to be - well, not much closer than we are. Relatively, that is. Roughly 0.975 AU from the star. To have a surface temperature of 281 K, it would have to be at 1.05 AU from the star. This neglects, by the way, the different levels of land/sea absorption.
So, actually, if the Earth's orbit had an eccentricity of about 0.35, it could have some moderate seasons.
I hope this helps.
Note: I used this eccentricity calculator for the sake of accuracy and time. I would still be obliged if anyone could check my overall calculations for this answer.