Vegetarianism is much more common than veganism in Cyprus but there is a small vegan community. Greeks may fast at certain times of the year for religious reasons which involves removing some or all animal products from the diet (some sea animals may be left out). If an item is indicated as being suitable for fasting there is a good chance it is vegan.
At least in the capitol, Nicosia, there are many health food and organic stores which have vegan goodies, including hygiene products. Items in the major supermarkets are usually only specifically labelled vegetarian/vegan if they are from English brands such as Tesco.
Bakeries are everywhere in Cyprus. At the smaller bakeries, where the chef is actually on site, you may request foods such as spanagopita (spinach pie) to be made for you from scratch without animal ingredients. Foods such as spanagopita, hortopita (greens pie), eliopita (olive pie), potatopita (potato pie), milopita (apple pie), and kelofkes (pumpkin pie) are ostensibly vegan but some bakeries may use non-vegan margarine or spread egg along the tops. If you are worried about bone-char processed sugar, sugar is usually only used in specifically sweet baked goods.
The chain bakery Pandora in particular lists the ingredients for its foods on its website in English and Greek. Other stores such as Debenhams and Lidl may also list their ingredients by their baked goods.
Delicious bean dishes (giant beans, lentil, okra, green beans, chickpeas, black eyed peas, green peas etc.) are common in the cuisine and you may find some other vegan dishes available at restaurants such as fava, vegetarian gubepia (vine leaves stuffed with rice and herbs), vegetarian yemista (peppers and tomatoes stuffed with rice and herbs), the famous Greek salad (hold the dairy feta), tost/kroutsinia (a hard crackerish bread which can be topped with olive oil, olive paste, tomatoes and oregano; hold the feta), and bulgar (make sure vegan).
Sutzouku is sold in some villages. It looks like candle sticks and is made by coating a string of almonds in a mixture of either carob syrup or epsima (grape syrup) and flour. Loukamadas (basically fried sweet dumplings) could usually be considered vegan except for the chance that their sugar is processed with bone-char. The honey served on top would also have to be subbed out with your own syrup (there are an abundance of syrups to choose from in Cyprus supermarkets). Similarly, amigthalata (sugary almond balls) are prepared from ground almonds, sugar and orange blossom water. You may also find carob chocolate selling in some health food stores- a naturally sweet (i.e. no need to add sugar) treat that tastes like a combination of caramel and chocolate. The pita is almost always vegan, except for a few brands which may have added sugar.
Since Cyprus is part of the EU, animal testing for cosmetics is illegal. You thus do not have to worry that your soap, toothpaste, sunscreen etc. were tested on animals.
Green olive oil soap which is sold at about a euro per bar is usually vegan and you can find it being sold in tourist destinations and through the Arkadi brand at the supermarket.
Tourists visiting Cyprus may be interested in items such as epsima (grape syrup), carob syrup, rose water, orange blossom water, olive oil, and olive paste found at Cypriot supermarkets.
In particular, it is a shame if you are in Cyprus that there is not a widely available vegan tzatziki as there is vegan mayonnaise. It can easily be made at home, preferably with a blender, by mixing 350g silken tofu (a light tasting brand such as Yutaka), 1 clove raw garlic, 1/2 teaspoon dry mint, 3.5 teaspoons lemon juice, 1.5 medium Greek cucumbers (well shredded and left to dry at least 20 minutes on a paper towel; equivalent to approx. 2/3 an English long cucumber), and 1/2 teaspoon salt. I have found no difference in taste between the soy and cow's milk varieties.