There are at least claims of having directly observed the orbitals of the hydrogen atom. (Link to full paper - ask your teacher or a grown up to help you in reading this full paper.) Whether these are true observations is more complicated and depends on your definition of observable.

Some consider orbitals to be either purely mathematical or as (part of) the state of the atom. (The state is sort of "the complete description and all information" about the system). In both of those cases, an orbital is not a thing that could ever be "seen" (or detected, measured, observed, whatever word you'd like to use); that is, they hold that an orbital is not a single quantity, so it cannot be measured.

Others recognize different definitions whereby making measurements that allow one to produce a picture of what an orbital was during an experiment means that the orbital has been observed. This is more the approach taken by authors who say they have observed orbitals.

Regardless of whether you consider orbitals to be observable or not, note that they are not the same shape as those taught in earlier science classes. Instead of electrons flying around nuclei as particles on well-defined paths like the planets around our Sun, electron orbitals are more like "clouds": vague, distributed regions around the nucleus where one is more likely to find a particular type of electron.

The table here shows the orbital shapes for many electron levels. The "solar system" type depiction of electrons comes from an early model of atoms and is convenient to look at but is not particularly accurate.