Finding an Ethernet transformer when I have it's characteristics
The manual you refer to, in the table 66, recommends the following part:

WE 7499211121A

This is RJ-45 connector with magnetics (transformer) built in, and Digikey says it has 1500 pieces available for shipping.

Related connection circuit is shown in fig. 68, with all required components listed in the table 66.


  I can't use an RJ-45 connector, due to lack of space. This is why I have to implement my own magnetics.


You must have stated it in the question.

Update: it seems datasheet you refer to is having a typo: PHY device P/N is TLK106 and not TKL106. And it is not clear if the OEM7 board is having this chip on the board or not apart from stating:


  The OEM7 uses the Texas Instruments TKL106 Ethernet PHY


and 


  The PHY layer is based on the Texas Instruments TKL106 Ethernet PHY.


Second statement assumes that TLK106 is on the board, and your task should be to just attach the components as shown on the figure 68.

You can use any transformer fitting the requirements, for example made by Pulse. This "marketing" sheet lists magnetics with some generic circuit at the last page, you must look into each individual magnetics datasheets to find out the match. For example - H2019NL.

I strongly advise prototyping first - with full diagnosis on the reliability and performance.