Small islands have been built in inland waters since prehistoric times, but using sea floor material to build an offshore island is a much larger project. As Peter Taylor and JBH have pointed out, it has been done twice off the coast of Dubai. Both those projects have been monitored for erosion and local environmental effects, some of which were significant. There is also Samphire Ho, which is not an island but a coastal extension of the British mainland built out of spoil from the Channel Tunnel.
The technology for such a project is clearly available but the volume of rock/sand/mud to be moved makes it extremely expensive in energy and money, and probably not financially sustainable in most cases. If extra land is the aim then Dutch-style sea walls, or large platforms on stilts or floats, would be cheaper.
The environmental impact would depend on the location, geology and the source of energy for the work. Any bottom-dwellers that couldn't swim out of the way would be destroyed, but the depressions and islands would provide a larger habitat with a range of depths, which might give a net increase in biodiversity. If the islands were simply piled-up sand, their footprints would be huge.
Any minerals exposed by the digging would mix with seawater, which might (temporarily) pollute a wider area with iron, silt or heavy metals.
The finished islands would affect waves, currents and the erosion patterns of any nearby coasts for as long as they lasted.