Many common rocket fuels can be created by biological processes. In the case of fossil fuels, they actually were.
Bipropellant fuel
Hydrogen is the most efficient rocket fuel by mass, which is often the most important factor for a spacecraft. It has several major drawbacks, though: it needs to be cooled down to ridiculously low temperatures, has an abysmal density meaning giant tanks, and has a tendency to boil off as well as to escape pretty much any container with time (H2 molecules are so small they slip between the tank's atoms). It can be produced by bacteria, or the civ could use biotech to reproduce today's industrial processes, which for many are relatively straightforward.
Kerosene (specifically highly purified RP1), on the other hand, is compact and doesn't need to be cooled down. It is quite a bit less efficient per mass than hydrogen, but those make up for it enough that it is a rather common rocket fuel. It comes from oil, which has naturally formed from dead organisms, and it is possible to recreate this process artificially from, say, algae. We don't do it today because it would be too expensive, but it may start to appear once we run out of fossil oil - nothing a biotech civ should have any problem with.
Between those, there is methane. Not as compact as kerosene, but not too bad either. Needs to be cooled down, but far from the extremes of hydrogen. Performances are also between those, as its hydrogen ratio in the molecule is higher than kerosene. Methane is a well-known organic byproduct.
Note that gaseous fuel is almost always cryogenically liquefied instead of compressed, as this allows greater density and doesn't require impractically heavy tanks.
For early rockets, ethanol can be used, and such a civ should have ample alcohol-producing capabilities. It is very easy to use as fuel compared to the others, but its poor performances mean that it will be only used for initial experimental rockets (think V2).
Bipropellant oxidiser
As oxidiser, liquid oxygen is the obvious choice. It is cryogenic (though not as extremely cold as hydrogen), but is the most efficient practical oxidiser.
If you hate the world and everything in it, you can use fluorine (aka burning cancer). The stuff is such a nightmare to handle even the Nazis thought it was too dangerous. It will burn anything it touches, produce highly toxic fumes and durably contaminate the zone. There are probably bacteria out there that can produce the stuff, but frankly the world is a scary enough place as it is.
Hydrogen peroxide is not as efficient and will try to explode when given the opportunity, but it is liquid at room temperature, compact and can be useful if you want a simpler, non-cryogenic rocket. Again, there are some organisms that produce it, so it shouldn't be a stretch for a biotech civ to produce it.
Nitrous oxide can also be produced biologically, and fills somewhat the same niche as hydrogen peroxide. You need to cool it somewhat, but not as much as other cryogenic fuels. It is a bit less compact as hydrogen peroxide, but is very stable.
Monopropellant
For things like satellites or small manoeuvring stages, they will want monopropellants, that is rocket fuel that doesn't need oxidisers. You only heat it and/or flow it against a catalyst and it will burn by itself.
Hydrogen peroxide and nitrous oxide can be used as monopropellants. Performances are mediocre compared to bipropellants, but small crafts will need small, simple engines and monopropellant engines are much simpler, and easier to be made tiny.
If you think Nazis were tree-hugging sissies, you can also explore chlorine trifluoride, which is the evil version of fluoride. It will burn water. It will burn sand. It will burn glass. It will burn spontaneously given the slightest provocation (for example, looking at it sideways). And of course its fumes will cancerously burn nearby lungs and contaminate the zone forever. But hey, at least it is efficient at it.