Vibration is your most likely and realistic answer.
Spider physiology provides very sensitive hairs on all of their legs. These hairs are extremely sensitive to vibration - this is how web-building spiders can detect prey entangled in the webs. If you have ever watched a spider catching a fly or other bug you are in for a fascinating lesson in vibration.
If you toss some bugs (or small leaves or tiny pebbles) into spider webs, and watch where the spider is lying in wait, you will first see a leg or two poke out and touch or grasp the web (some spiders have small claws at the tips of their legs). They will just sit there for a while, listening or rather feeling for vibrations of struggling prey. If they feel nothing, they will actually jerk the web strands, inducing sway and vibration into the web.
Then they will sit there some more listening, err, feeling the dynamic movement of the web in order to tell if something has been added. If it is really light (leaf) they will usually ignore it, unless there is a breeze, and it is flapping around against the web. If it is not so light (a bug, tiny stick or pebble), then they will venture out and take a gander at it. If it looks promising, or looks like it has damaged the web, then they will carefully approach and web it or cut it free. Oddly enough, they will web up and drink a ball of spit....
If a spider is large enough, as yours seem to be, it would have to deal with the macro scale more than the micro scale, and it would be possible for more advanced forms of vibration sensing to come into play. For example, echolocation. Humans are (somewhat surprisingly) capable of learning echolocation, it would be rather less surprising in a biology devoted to sensing vibration.
If they could send out detectable pulses, rather than a passive sense, then they could communicate using a more complex version of Morse code.
Let's give a nod to click-speak used by Julie E. Czerneda's Ganthor (one of Esen's favorite races) and many other civilized species who lack the appropriate mouth (or other) parts necessary to produce sounds intelligible to each other. A percussive language best done with the clicking of hooves (and the occasional stamp for emphasis) which is somewhat simplistic in grammar, but more than enough for basic mercantile or mercenary exchange along with other important basics.
Music, the universal language... now featuring Spyyyyy-dar!
Personally, I really like the idea of using strands of web as string instruments and having a musical based language which was suggested elsewhere. My mind is already going Battle of the Bands where spiders use different forms of music to convey not only words and meaning, but emotions as well. Blues with a touch of hard rock for that melancholy laced with anger, and smooth jazz interspersed with some dubstep for political commentary.
You know, spiders would be really awesome at dubstep... just visualizing it is putting my imagination into overdrive.
Taste and Smell
It is also worth mentioning that a spiders sense of taste and smell is mostly located in the legs as well. Those hairs are very sensitive and multi-function. Should they be able to scent or leave tastes upon web stands, you could have a very interesting and complex form of non-verbal, non-symbol based communication.
Telepathy, no really, hear me out here
If you are willing to get creative with biology, an RFID-like pseudo-psionic ability to communicate is also theoretically possible.
It turns out that the network of blood vessels and cerebro-spinal fluid channels in the human brain actually function as a natural Faraday cage, blocking out external electromagnetic signals. It is theorized that some people, those who claim to hear voices or other signals with their brain have gaps in the natural Faraday cage, which allows external electromagnetic signals to stimulate neuron clusters, which give rise to anomalous brain activity which is interpreted by the person in question as psychic phenomena and so forth.
Quantum paired particles shared between people is also another proposed possible source of this.
In any case, should you be willing to get creative with biology, then a species of creatures with natural gaps in the sheathing around the brain, may actually be able to exchange recognizable signals between pairs of creatures who are close enough (about the range of an RFID tag), especially if they have developed actual signal transmitting organs.
Along with this idea, I should like to mention the excellent light novel Mother of Learning, which has some GIFTS as major characters in the story (Giant Intelligent Friendly Talking Spiders) who use more traditional psionics ala telepathy to communicate with those few humans capable of doing so. They have some interesting insights into what sentient spiders would be like, so worth a reference.
Web spinning sentients may also devise a written form of communication using their built in means: webs. A spider holding up a Cat's Cradle between it's forelegs may indicate a desire to talk, or perhaps a merchant with some interesting wares (pickled giant flies, anyone, fresh from the vat!)
And let us not forget Arthur C. Clark's (iirc) contribution: spiders with the ability to control the colors on their skin, who "talk" via bands of color.