There isn't enough information to give a good answer.

The simple answer is that impedance matching is needed if a component or element in the system is not already matched to a specified impedance.

TI's documentation implies the IC is already 50 ohms at the package, but they do not provide measured impedance data to quantify what the exact impedance is over frequency. The documentation also specifies that all the transmission lines on the application board are to be 50 ohms.

If all this is true then you need to know the antenna impedance and the impedance of any filters and/or baluns used to interface the part to the antenna. Matching components are often needed to optimize the insertion loss and return loss performance of a balun or filter.

Finally there is the antenna. The small dual-band antenna you intend to use is matched to 50ohms only over a small frequency range for the 2.4GHz band. The 5GHz band match is more broadband by comparison. The match is specified through the VSWR performance. A VSWR of 2:1 is generally an acceptable worst case impedance mismatch (this is a return loss of ~10dB). The antenna  VSWR is less than 2:1 over the range of 2.42 - 2.46 GHz, so you have about 40MHz to operate where the match and efficiency are good. If you need to operate at other frequencies you will need a matching network and the efficiency will not be very good. This is the price you pay to use an ultra compact antenna. Bigger antennas will generally provide wider bandwidth performance; although, matching networks are also generally required to provide a desired impedance over a frequency range.

Are you just trying to use the TI evaluation board to interface to the two antennas, or are you trying to design your own board that will interface to the two antennas?

WL1835MODCOM8B:
WiLink™ 8 Module 2.4 GHz WiFi + Bluetooth COM8 Kit for Sitara

The Sitara module card has a printed antenna plus a miniature coax output (probably a smt jumper).

Unfortunately you have a large task ahead of you because in addition to finding out the IC impedance, antenna impedance and designing matching networks (if needed) you also need to make sure the transmission lines for your PCB design are the correct impedance.

All the tasks can be done, but it will be quite a learning experience for you if you are responsible for making sure it all works correctly. It is good practical experience, but it will be a challenge if you don't have equipment and/or simulation tools to help you through the design and evaluation phase.

Unfortunately, I don't have a good suggestion for a cheap VNA; although, your plan for renting one is reasonable.

More importantly you need
1. to find out from TI what the IC impedance is over frequency

2. figure out how to design a 50 ohm microstrip transmission line using the PCB stackup of your vendor

3. find out the impedance of the baluns and other components (switches) you plan to use

4. develop matching networks if necessary to keep the system port impedances matched

5. write down a measurement test plan of how you are going to verify the matches are as expected using you PCB. You may find you want to design you board to make it easier to conduct your measurement plan. For instance, you may want a few test structures like transmission lines of various lengths that you can measure separately from the IC circuit. This would help you verify that your transmission lines impedances are close to the design targets.

6. if matching networks are necessary then you will need some sample kits of inductors and capacitors to use to tweak the match designs after assembly to optimize performance.

Of course you'll need to learn the basics of impedance matching and microstrip transmission line design. All of this is classic information, so you should be able to pick it up and apply it fairly quickly.