How exactly to implement the fractional part of a 'fractional equalizer'?
A typical ("non-fractional") equalizer operates at the symbol rate. That is, the inputs that you apply to the equalizer are generally soft decisions on what the unequalized received symbols are. A linear equalizer then takes this stream of soft symbol metrics and applies a linear filter that (hopefully) corrects for any non-ideal frequency response found in the path from modulator to demodulator.

The only difference with a fractionally-spaced equalizer (FSE) is the sample rate that it runs at: instead of the one sample per symbol that you see with a "standard" equalizer, an FSE operates at some multiple of the symbol rate. Typical values I've seen before are $T/2$- and $T/4$-spaced equalizers, which operate at twice and four times the symbol rate, respectively. 

These samples should be straightforward to obtain; whatever processing steps in your system are used to generate the symbol decisions (e.g. when obtaining symbol timing synchronization) almost certainly operate at a minimum of 2 samples per symbol. Instead of decimating the receiver output to one sample per symbol, keep the higher sample rate intact through the equalizer in that case.