Yep, we could in theory.
What we'd need, as you say, is some giant reflector placed at a Lagrange point L1 between the Earth and Sun. At this point the gravitational force on the satellite from the Earth and Sun cancel. The satellite would also appear 'fixed' in the sky (relative to the Earth, ignoring the Earth's spin) as the Earth orbits the Sun and would be about 1 million miles from Earth.
The problem is the size. Lawrence Livermore National Laboratory did some calculations and found to block about 1 to 2% of the Sun's light from reaching Earth, the mirror would need to be 600,000 square miles in area. That's about the size of Greenland. Researchers at the University of Arizona suggested using lots of smaller mirrors instead. The cost of this solution, however, was 26 times the US national debt.
Another option that was looked at by a consultancy group Star Technology was to use a series of steerable mirrors around the equator. However, you'd need about 5 million satellites and, based on estimated failure rates, you'd be replacing or repairing around 140 satellites a day.
So the physics says yes, we could do this. The engineering says no, not yet, but possibly in the future. And the economics says don't even try!
Additional - Just to make good on the comment ... the Earth is about 7918 miles in diameter, so if you take the 'shadow area' of the hemisphere facing the sun you get an area of PI x 3959^2 which is about 49,240,321 square miles. Since the sun is 93 million miles away, you can assume that the rays reaching this 'shadow area' are nearly parallel rays. So if you want to block or reflect 1.5% of the sunlight, your space mirror or sun-blocker needs to have an area of 1.5% of 49,240,321 = 738,604 square miles. This would be a circular mirror/blocker with a diameter of about 970 miles, or a square mirror/blocker about 860 miles on each side!
What's even more interesting is that if you take that the solar flux is about 1360 Watts per square meter, and you used a 99% reflective mylar panel, the mirror would be absorbing 13.6 Watts per square meter or a total of about 10 million watts of energy! How that affects the mirror (temperature and expansion) is a calculation you could do.
But if you had a mirror that big, the question I'd have is how would solar winds affect its position?