Frank Borman is about to get new competition on some routes flown by his Eastern Air Line jets and from an airline that also has a former astronaut as a top executive.

Donn Eisele, an astronaut on the Apollo 7 flight to the moon, joined Washington businessman O. Roy Chalk's new Trans Carib Air last month as executive vice president -- the man who will be in charge of all flight operations. Starting in May or June, Trans Carib hopes to earn its wings every day with flights between New York and San Juan, Puerto Rico. By mid-1980, Trans Carib expects to have its first big plane -- a Lockheed L-1011-200 TriStar -- available for the Caribbean service.

Chalk, the flamboyant former chairman of Trans Caribbean Airways and the D.C. Transit System, signed an agreement with Lockheed Corp. this week to buy one of the TriStars as well as an option for two additional planes. The three aircraft would cost more than $100 million.

According to Eisele, Trans Carib initially will operate only on the New York-San Juan route but plans for the near future include a San Juan-Miami route and a Miami-Houston-California route, with San Juan and the Virgin Islands as destination points.

"Further expansion is expected to include additional East Coast points of origin such as Boston, Baltimore and Washington as well as additional destination points in the Caribbean, including the Virgin Islands," the former astronaut stated.

He said Trans Carib hopes to offer "all coach" and "all first class" flights at low fares. Trans Carib flights to San Juan were approved by the Civil Aeronautics Board on Dec. 19 with one-way day coach fares as low as $79 and night-time fares of $69, compared with more than $100 on coach flights today by Eastern and American Airlines.

Reflecting the growing age of mass airline transportation, Trans Carib won't sell tickets direct to the public, except on a space available basis two hours before departure -- when the fare will be $49. All other tickets will be sold through travel agencies.

The new Chalk airline has many of the characteristics of the old Chalk airline, Trans Caribbean, which he formed after World War II with surplus military aircraft. That company was merged into American Airlines in 1971, but Chalk's agreement did not include a promise to stay out of the business.

Chalk chartered Trans Carib last year as a private firm not affiliated with other Chalk enterprises. In a telephone interview from Palm Beach, Fla., he said it's "quite exciting" to be an airline chief again and he vowed to attract one-third of the traffic on the New York-San Juan route -- roughly the same share he had with his former line, in competition then with Eastern and Pan American World Airways.

Chalk said yesterday that Trans Carib will lease wide-bodied or conventional jets to begin service within a few months. Private bank and institutional lending has been arranged to finance the airplane leases and purchases, Chalk added.

Claiming that his old airline had pioneered some of the no-frills mass transportation concepts attributed in recent years to Britain's Freddie Laker, Chalk promised to specialize in a low-cost tourist market and service to New York's large Puerto Rican community.

Trans Carib is the fourth airline to buy the TriStar-200 models. The others are Saudi Arabian Airlines, Delta Air Lines and British Airways. Lockheed officials said the first TriStar would be delivered to Chalk in June 1980 with the options scheduled for 1981.