The short answer to your question is yes. But, the best answer to your question is maybe.  First of all, it is important to know that vascular problems in general whether due to blockage from plaque or due to a clot have inflammation as a common denominator.  Whether you’re talking about a coronary artery or a carotid artery or an intracranial artery its arterial inflammation that sets the stage for the problem.  By way of example, its arterial inflammation that acts as the “Velcro effect” enabling cholesterol plaque to build up.  As it turns out any repository of inflammation in the body can contribute to this “Velcro effect”.  Thus, chronic sinusitis has this potential.  But what is important to understand is that its untreated, smoldering chronic sinusitis that has this potential. 

There is some evidence that the proximity of the sinus inflammation to the carotid arteries and the intracranial arteries gives this a more potent negative effect in terms of stroke initiation. 

In your particular case you treat the chronic inflammation with your allergy shots, your Singular (a non-steroid respiratory anti-inflammatory) and your topical nasal steroid spray.  However, some people choose not to treat a chronic sinusitis and therein lies the potential mischief. 

The worst case I have personally seen was a patient of mine who had chronic infected sinuses producing yellow and green mucus who declined therapy from me, her PCP and an ENT.  She suffered both a stroke and a brain abscess from the condition.

Another way that chronic sinusitis might lead to stroke is due to self-medication with either topical or oral decongestants.  Both have the potential to raise blood pressure and pulse and

if used as a chronic therapy (as opposed to brief and occasional use) they can add to the hazard of stroke.