The answer is yes; according to our current theoretical ideas and observations of circumstellar disk lifetimes, planets must be able to form prior to the ignition of the main phase of hydrogen burning in stars on the main sequence with masses similar to, or less than, the Sun.
The Details:
I take your question to mean planet formation prior to the fusion of hydrogen into helium, not the very brief phase during which the star's primordial deuterium is burned, which for a star like the Sun happens within the first million years and certainly will be initiated before planets can fully form.
Planets form in a disk of circumstellar material around their parent protostars. The "core-accretion" model of giant planet formation suggests that it takes only 5-10 million years to form a giant planet in this disk. The main competing model (thermal instability in the disk) suggests an even more rapid formation timescale. Stars of a solar mass or less take significantly longer than this to contract sufficiently that their cores reach the necessary temperatures for hydrogen ignition.
To put some numbers on this: according to the pre-main sequence models of Siess, Dufour & Forestini (2000), it takes about 25 million years for a solar mass star to produce an appreciable fraction of its luminosity from hydrogen burning. This timescale gets longer for lower mass stars.
Statistically speaking, we know that most stars lose their disks within about 10 million years (see for example Hillenbrand 2008). They are essentially gone by an age of 25 million years, so all giant planet formation, which relies on gass-accretion for the disk, must have already occurred by then. However, we also now know thanks to extensive surveys for exoplanets that a large fraction of stars have gas giant planets. So most stars must form planets within about 5-10 million years and this is definitely not enough time for H-burning to start in a star of $\leq 1M_{\odot}$. The same argument probably does not apply to small rocky planets, which probably take a bit longer to reach their final configuration (maybe as long as 100 million years in our Solar System), although large planetesimals should be present after only a million years.