Can I make a planet out of mud?

$10^{26}$ kg is about 5% of the mass of Jupiter and much larger than the mass of the Earth. Gravitational contraction would ensure that your mass of "mud" would indeed form a (roughly) spherical planet. The timescale upon which it does so is entirely dependent on what sort of radius you began with. The "dynamical" freefall timescale is given approximately by $(G\rho)^{-1/2}$, where $\rho$ is the average density.

Hence if the initial mass of $10^{26}$ kg was spread over an astronomical unit in diameter, then it would take about 15 years. However, this timescale is a lower limit to the time it takes to settle into an equilibrium configuration, since it assumes the material can fall freely. As it becomes more compact, the material will heat up considerably - according to the virial theorem about half the released gravitational potential will be radiated away (largely in the infrared) and half will heat the material up. The interior will become a gas/fluid and exert a pressure that will slow the contraction.

Whether the contracting mass can then initiate nuclear fusion and become a star depends on a competition between whether the temperature can become hot enough to allow nuclei to have enough kinetic energy to approach each other closely enough to interact via the strong nuclear force (although quantum mechanical tunneling is of vital importance here) and whether the density becomes so high that the material is supported by electron degeneracy, whereby the Pauli exclusion principle demands that electrons in the gas occupy different quantum states and results in an almost temperature-independent pressure that prevents further contraction. In the latter case, the "planet" could simply cool-off at roughly constant size and nuclear fusion would never occur.

By convention, "a star" is defined as a body which can initiate the fusion of hydrogen (protons). Assuming your mud contains water and hence plenty of hydrogen, then in a pure hydrogen gas, the threshold mass at which gravity wins this particular battle and the gas heats up sufficiently to start hydrogen fusion is around 75 Jupiter masses. But "mud" also contains a mixture of carbon, silicon and oxygen. In this case the threshold will be much higher because the Coulomb barrier between a pair of carbon nuclei is much higher than between a pair of hydrogen nuclei. Roughly speaking, a mass of a bit more than the mass of the Sun is capable of being supported by electron degeneracy alone before carbon fusion can occur.

Your mud is somewhere between these two extremes - it has hydrogen to fuse, but most of its mass is carbon, oxygen and silicon. In any case $10^{26}$ kg is too low by orders of magnitude to become a star. There is a lower threshold at about 13 times the mass of Jupiter where the deuterium (a trace isotope of hydrogen) would fuse at lower temperatures than hydrogen. However, this is not classed as "a star" since the entire deuterium content is burned in (astrophysically-speaking) a short time period of a few tens of millions of years.

The entire life history (  path on the HR-diagram https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram  ) of a star is determined by its initial mass, chemical composition, and angular momentum. Too much angular momentum and all your initial mass goes spinning off into space. But "float freely" sounds like no angular momentum. So, depending on your definition of "planet", you'll certainly see any such initial mass contract, and $10^{26}\mbox{kg}$ sounds like enough to make it pretty much spherical.

But even if your "mud" were pure hydrogen, that's not enough mass such that gravitational contraction would heat it up enough to "ignite". So no star. But some off-main-sequence stars can eventually burn carbon (  https://en.wikipedia.org/wiki/Carbon-burning_process  ), so if you bumped up your mass enough, and if "mud" is carbon plus some trace elements, it might just start off with that carbon process. But I don't offhand know what initial mass (of carbon) would be required for that. (For an initial mass of hydrogen, and if memory serves, you need about ten Jovian masses for ignition, which is $\sim2\times10^{28}\mbox{kg}$.)

If you take all your mud in one trip, then the planet will start to form inside your vehicle and by the time you find the place, it already close to being a planet.

But if you do it in multiple/numerous small trips, then, even if you find a place to float freely, you have to know how the first dump is moving wrt your big planet so that you can locate it when you bring over the second load.

Suppose you are able to dump all the trips pretty close to one another.

Even in this case, it seems, by the time you are done dumping, it should already be very close to be a planet because the formation will not wait for you to finish dumping.

As others have said, it will be a watery planet. You better make it orbit in habitable zone of some mid age sun like star.

You will not ever see a star from this. No calculations are needed here. Because, you are digging a single "big planet" to make a new planet. That says it all.