Have we ever placed an object orbiting our Sun and the Sun only? like a breadcrumb of our orbit...
I was reading about VLA for satellites and wouldn't this breadcrumb be a viable solution to create a massive VLA?
Sure. My initial thought was to look in particular at solar observatories, because I know that several have been moved away from the Earth. Wikipedia has a list.
The short version seems to be
It is also true that the interplanetary probes and asteroid and comet missions are in solar orbit during their cross-trajectories (well, a few have been on escape trajectories if you don't like that).
I don't quite know what you mean by "breadcrumb", but this NASA page talks about satellites at Earth's L1 and L2 points, where they follow Earth's orbit around the Sun precisely without actually orbiting the Earth. Unfortunately these points are "unstable equilibria" and are a little like balancing a broom on its end in your hand: you have to actively correct for the system's natural tendency to "fall out of orbit". Partly to help handle this -- but mostly because you don't want to point your satellite dish directly at the Sun while detecting a weak satellite's scientific data -- usually those spacecraft are orbiting the actual L1/L2 point in a so-called "halo orbit" (since at the L1 point it describes a halo about the Sun while seen from Earth), so it's not orbiting "the sun only."
Indeed, you want something which is orbiting "with Earth but beside it", but unfortunately these objects will typically have a net acceleration causing them to fall in towards Earth. However, there are two other such Lagrange points, L4 and L5, where gravity balances out "on the orbit", and these are stable.
Unfortunately, since they're stable, they tend to collect rocks and other space debris from our orbit! I don't know that anybody has actually stationed a semipermanent artificial satellite in either L4 or L5. Some Googling mentions the NASA STEREO mission which was (is?) underway to visit those points and see what they've got in them.