Does the entropy of a chemical system increase in an endothermic reaction?
Not necessarily.
Exo- and endothermic refer to the direction in which heat is exchanged between system and surroundings, not to changes in the entropy of the system, or of the temperature of either system or surroundings. Your logic though is partly correct: exothermic means the entropy of the surroundings increases, as it would do with an increase in T - the catch is that we usually assume that the surroundings has constant T, but that's because we say it has infinite heat capacity.
It is a recurrent subject of confusion to conflate changes in the entropy of the surroundings, of the system, and of the sum of both. Always keep the second law in mind. What determines whether a reaction will be spontaneous is the sum of the changes in entropy of the surroundings and the system.
When a reaction occurs reversibly, $\Delta S_{\mathrm{surr}} + \Delta S_{\mathrm{sys}}=0$.
When spontaneous, $\Delta S_{\mathrm{surr}} + \Delta S_{\mathrm{sys}}&gt;0$.
When a reaction is exothermic,  $\Delta S_{\mathrm{surr}} &gt;0$.
When a reaction is endothermic,  $\Delta S_{\mathrm{surr}} &lt;0$.
Nothing about the thermicity (whether the reaction is endo or exothermic) tells you what happens to $\Delta S_{\mathrm{sys}}$, unless you already know whether the reaction is spontaneous or not (in which case the sum of the entropy changes is positive).