JOEL - I know of no official counsel from church leaders on this matter. The Church General Handbook states the following:
"Brethren should administer to the sick at the request of the sick person or of someone who is vitally concerned so the blessing will be according to their faith (see D&C 24:13-14; 42:43-44, 48-52)."
Animals cannot of course exercise faith the way humans do; but there is the faith of the owner of the pet that should count for something.
Marion G. Romney said:
"Priesthood is power, the power of God, a vital source of eternal strength and energy delegated to men to act in all things for the well-being of mankind, both in the world and out of it (DS 3:80; Romney, p. 43)."
The blessing of a pet might be considered acting "in all things for the well-being" of the owner if they are sincerely concerned and emotionally upset about it.
As you said pioneers used the priesthood to bless sick oxen that they needed to survive the journey as told in the following story:
"As the company was crossing Wyoming one day, one of Mary’s oxen suddenly lay down as if poisoned. It appeared the ox would die, and Mary had no spare ox with which to replace him. As the ox began to stiffen, the company captain exclaimed, “He is dead, there is no use working with him, we’ll have to fix up some way to take the Widow [Mary] along. I told her she would be a burden on the company.”
Mary said nothing, but she took a bottle of consecrated oil from her wagon and asked her brother, Joseph Fielding, and another man to administer to her ox. “It was a solemn moment there under the open sky. A hush fell over the scene. The men removed their hats. All bowed their heads as Joseph Fielding … laid his hands on the head of the [dying] ox, and prayed over it. The great beast lay stretched out and very still. Its glassy eyes looked nowhere. A moment after the administration the animal stirred. Its huge, hind legs commenced to gather under it. Its haunches started to rise. The forelegs strengthened. The ox stood and, without urging, started off as if nothing had happened.” Soon another ox fell ill and was administered to, and it also recovered."(Life of Joseph F. Smith)

I guess if it worked for their oxen it could work for any other animal that is important to a person. If you feel uncomfortable using oil, the blessing could be more in the form of a prayer, calling upon Heavenly father and invoking priesthood power; much the same way a new baby is blessed. Much of what you would do in this sort of situation would depend greatly on inspiration from the Holy Ghost. But I do not think God would be offended, and it can serve as a comfort to the owner of the pet.

CHAD - Can you give a blessing with annointing without a companion?

JOEL - According to the Church General Handbook, refering to anointing with oil and administering to the sick:
"Only Melchizedek Priesthood holders may administer to the sick or afflicted. Two or more of them normally do it together, but one may do it himself.
The anointing is done by one Melchizedek Priesthood holder."
For sealing the anointing: "Normally, two or more Melchizedek Priesthood holders place their hands lightly on the head of the person." (Page 30)
As you can see it is appropriate for only one to do it all in an emergency if there is no other priesthood holder available. But if at all possible it is better to have two Melchizedek priesthood holders for administering to the sick because, as in all other priesthood ordinance work, two witnesses are always better than one; “in the mouth of two or three witnesses shall every word be established” (2 Cor. 13:1; see also Deut. 17:6; Deut. 19:15; Matt. 18:15-16; John 8:12-29). The extra witness confirms that the event took place and that the ordinance is done corrrectly.
According to President Kimbal:
"Sometimes when oil is not available, or two brethren are not present, or when the sick one has recently been anointed, a substitute program is followed whereby one or more elders give a blessing, likewise in the name of the Lord and by authority of the Melchizedek Priesthood. He will pronounce such blessings as seem appropriate and as the Spirit moves." (New Era, Oct. 1981, 45)
Normally, what I do if I am alone is to just give the person a priesthood blessing of health without using the oil, as President Kimbal describes. This is just as effective and appropriate as doing it with the oil, because it is not the oil that heals the person but the priesthood power and faith of those invloved.